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  <meta name="description" content="摘要：奇异值分解（singular value decomposition）是线性代数中一种重要的矩阵分解，在生物信息学、信号处理、金融学、统计学等领域有重要应用，SVD都是提取信息的强度工具。在机器学习领域，很多应用与奇异值都有关系，比如推荐系统、数据压缩（以图像压缩为代表）、搜索引擎语义层次检索的LSI等等。（本文原创，转载必须注明出处.）">
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<meta property="og:description" content="摘要：奇异值分解（singular value decomposition）是线性代数中一种重要的矩阵分解，在生物信息学、信号处理、金融学、统计学等领域有重要应用，SVD都是提取信息的强度工具。在机器学习领域，很多应用与奇异值都有关系，比如推荐系统、数据压缩（以图像压缩为代表）、搜索引擎语义层次检索的LSI等等。（本文原创，转载必须注明出处.）">
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<meta name="twitter:description" content="摘要：奇异值分解（singular value decomposition）是线性代数中一种重要的矩阵分解，在生物信息学、信号处理、金融学、统计学等领域有重要应用，SVD都是提取信息的强度工具。在机器学习领域，很多应用与奇异值都有关系，比如推荐系统、数据压缩（以图像压缩为代表）、搜索引擎语义层次检索的LSI等等。（本文原创，转载必须注明出处.）">
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          <h2 class="post-title" itemprop="name headline">一步步教你轻松学奇异值分解SVD降维算法
              
            
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<p>摘要：奇异值分解（singular value decomposition）是线性代数中一种重要的矩阵分解，在生物信息学、信号处理、金融学、统计学等领域有重要应用，SVD都是提取信息的强度工具。在机器学习领域，很多应用与奇异值都有关系，比如推荐系统、数据压缩（以图像压缩为代表）、搜索引擎语义层次检索的LSI等等。（本文原创，转载必须注明出处.）</p>
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<h1 id="奇异值分解原理"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3%E5%8E%9F%E7%90%86" class="headerlink" title="奇异值分解原理"></a>奇异值分解原理</h1><h2 id="什么是奇异值分解-SVD）"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E4%BB%80%E4%B9%88%E6%98%AF%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3-SVD%EF%BC%89" class="headerlink" title="什么是奇异值分解(SVD）"></a>什么是奇异值分解(SVD）</h2><blockquote>
<p>奇异值分解</p>
</blockquote>
<p>假设M是一个m×n阶矩阵，其中的元素全部属于域K，也就是实数域或复数域。如此则存在一个分解使得</p>
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font-family: MathJax_Main;">×</span><span class="mi" id="MathJax-Span-9" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-10" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="msubsup" id="MathJax-Span-11" style="padding-left: 0.301em;"><span style="display: inline-block; position: relative; width: 2.562em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.78em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-12" style="font-family: MathJax_Math-italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.807em; left: 0.658em;"><span class="texatom" id="MathJax-Span-13"><span class="mrow" id="MathJax-Span-14"><span class="mi" id="MathJax-Span-15" style="font-size: 70.7%; font-family: MathJax_Math-italic;">m</span><span class="mo" id="MathJax-Span-16" style="font-size: 70.7%; font-family: MathJax_Main;">×</span><span class="mi" id="MathJax-Span-17" style="font-size: 70.7%; font-family: MathJax_Math-italic;">m</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-18"><span style="display: inline-block; position: relative; width: 2.384em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.66em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-19" style="font-family: MathJax_Main;">Σ</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.807em; left: 0.717em;"><span class="texatom" id="MathJax-Span-20"><span class="mrow" id="MathJax-Span-21"><span class="mi" id="MathJax-Span-22" style="font-size: 70.7%; font-family: MathJax_Math-italic;">m</span><span class="mo" id="MathJax-Span-23" style="font-size: 70.7%; font-family: MathJax_Main;">×</span><span class="mi" id="MathJax-Span-24" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-25"><span style="display: inline-block; position: relative; width: 2.086em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.78em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-26" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.182em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.336em, 1000.6em, 4.17em, -999.997em); top: -4.342em; left: 0.896em;"><span class="mi" id="MathJax-Span-27" style="font-size: 70.7%; font-family: MathJax_Math-italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.455em, 1001.49em, 4.17em, -999.997em); top: -3.807em; left: 0.598em;"><span class="texatom" id="MathJax-Span-28"><span class="mrow" id="MathJax-Span-29"><span class="mi" id="MathJax-Span-30" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span><span class="mo" id="MathJax-Span-31" style="font-size: 70.7%; font-family: MathJax_Main;">×</span><span class="mi" id="MathJax-Span-32" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.361em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>M</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msub><mo>=</mo><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mo>×</mo><mi>m</mi></mrow></msub><msub><mi mathvariant="normal">Σ</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msub><msubsup><mi>V</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>×</mo><mi>n</mi></mrow><mi>T</mi></msubsup></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-1">M_{m×n}=U_{m×m} \Sigma_{m×n} V^T_{n×n}</script><p class=" has-jax has-jax">其中U是m×m阶酉矩阵；Σ是m×n阶非负实数对角矩阵；而<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-2-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-33" style="width: 1.729em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.432em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.61em, 1001.43em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-34"><span class="msubsup" id="MathJax-Span-35"><span style="display: inline-block; position: relative; width: 1.432em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.78em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-36" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.182em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.342em; left: 0.896em;"><span class="mi" id="MathJax-Span-37" style="font-size: 70.7%; font-family: MathJax_Math-italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>V</mi><mi>T</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-2">V^T</script>，即V的共轭转置，是n×n阶酉矩阵。这样的分解就称作M的奇异值分解。Σ对角线上的元素<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-3-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x3A3;&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-38" style="width: 1.253em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.789em, 1001.01em, 2.979em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-39"><span class="msubsup" id="MathJax-Span-40"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.66em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-41"><span class="mrow" id="MathJax-Span-42"><span class="mo" id="MathJax-Span-43" style="font-family: MathJax_Main;">Σ</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.807em; left: 0.717em;"><span class="mi" id="MathJax-Span-44" style="font-size: 70.7%; font-family: MathJax_Math-italic;">i</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.146em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>Σ</mo></mrow><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-3">Σ_i</script>,i即为M的奇异值。常见的做法是将奇异值由大而小排列。如此Σ便能由M唯一确定了。（虽然U和V仍然不能确定。）</p>
<ul>
<li class=" has-jax has-jax">V的列组成一套对<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-4-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-45" style="width: 1.253em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.789em, 1001.01em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-46"><span class="mi" id="MathJax-Span-47" style="font-family: MathJax_Math-italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.063em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 0.932em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></span></span><script type="math/tex" id="MathJax-Element-4">M</script>的正交”输入”或”分析”的基向量。这些向量是<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-5-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;&amp;#x2217;&lt;/mo&gt;&lt;/msup&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-48" style="width: 3.098em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.562em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1002.56em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-49"><span class="msubsup" id="MathJax-Span-50"><span style="display: inline-block; position: relative; width: 1.551em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-51" style="font-family: MathJax_Math-italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.342em; left: 1.134em;"><span class="mo" id="MathJax-Span-52" style="font-size: 70.7%; font-family: MathJax_Main;">∗</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mi" id="MathJax-Span-53" style="font-family: MathJax_Math-italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.063em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>M</mi><mo>∗</mo></msup><mi>M</mi></math></span></span><script type="math/tex" id="MathJax-Element-5">M^*M</script>的特征向量。</li>
<li class=" has-jax has-jax">U的列组成一套对<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-6-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-54" style="width: 1.253em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.789em, 1001.01em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-55"><span class="mi" id="MathJax-Span-56" style="font-family: MathJax_Math-italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.063em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 0.932em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></span></span><script type="math/tex" id="MathJax-Element-6">M</script>的正交”输出”的基向量。这些向量是<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-7-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;&amp;#x2217;&lt;/mo&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-57" style="width: 3.098em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.562em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1002.56em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-58"><span class="mi" id="MathJax-Span-59" style="font-family: MathJax_Math-italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span class="msubsup" id="MathJax-Span-60"><span style="display: inline-block; position: relative; width: 1.551em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-61" style="font-family: MathJax_Math-italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.342em; left: 1.134em;"><span class="mo" id="MathJax-Span-62" style="font-size: 70.7%; font-family: MathJax_Main;">∗</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><msup><mi>M</mi><mo>∗</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-7">MM^*</script>的特征向量。</li>
<li class=" has-jax has-jax">Σ对角线上的元素是奇异值，可视为是在输入与输出间进行的标量的”膨胀控制”。这些是<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-8-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;&amp;#x2217;&lt;/mo&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-63" style="width: 3.098em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.562em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1002.56em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-64"><span class="mi" id="MathJax-Span-65" style="font-family: MathJax_Math-italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span class="msubsup" id="MathJax-Span-66"><span style="display: inline-block; position: relative; width: 1.551em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-67" style="font-family: MathJax_Math-italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.342em; left: 1.134em;"><span class="mo" id="MathJax-Span-68" style="font-size: 70.7%; font-family: MathJax_Main;">∗</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><msup><mi>M</mi><mo>∗</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-8"> MM^* </script>及 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-9-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;&amp;#x2217;&lt;/mo&gt;&lt;/msup&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-69" style="width: 3.098em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.562em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1002.56em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-70"><span class="msubsup" id="MathJax-Span-71"><span style="display: inline-block; position: relative; width: 1.551em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-72" style="font-family: MathJax_Math-italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.342em; left: 1.134em;"><span class="mo" id="MathJax-Span-73" style="font-size: 70.7%; font-family: MathJax_Main;">∗</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mi" id="MathJax-Span-74" style="font-family: MathJax_Math-italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.063em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>M</mi><mo>∗</mo></msup><mi>M</mi></math></span></span><script type="math/tex" id="MathJax-Element-9"> M^* M </script>的特征值的非负平方根，并与U和V的行向量相对应。</li>
</ul>
<h2 id="矩阵知识"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E7%9F%A9%E9%98%B5%E7%9F%A5%E8%AF%86" class="headerlink" title="矩阵知识"></a>矩阵知识</h2><blockquote>
<p>正交与正定矩阵</p>
</blockquote>
<ul>
<li>正交矩阵：若一个方阵其行与列皆为正交的单位向量，则该矩阵为正交矩阵，且该矩阵的转置和其逆相等。两个向量正交的意思是两个向量的内积为 0。 <a href="https://zh.wikipedia.org/wiki/%E6%AD%A3%E4%BA%A4%E7%9F%A9%E9%98%B5" target="_blank" rel="noopener">&gt;&gt; 正交矩阵知识扩展</a></li>
<li class=" has-jax has-jax has-jax">正定矩阵：如果对于所有的非零实系数向量 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-10-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-75" style="width: 0.598em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(2.027em, 1000.48em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-76"><span class="mi" id="MathJax-Span-77" style="font-family: MathJax_Math-italic;">z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 0.718em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math></span></span><script type="math/tex" id="MathJax-Element-10">z</script>，都有 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-11-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/msup&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;&amp;gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-78" style="width: 5.003em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.17em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.61em, 1004.11em, 2.86em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-79"><span class="msubsup" id="MathJax-Span-80"><span style="display: inline-block; position: relative; width: 1.074em; height: 0px;"><span style="position: absolute; clip: rect(3.396em, 1000.48em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-81" style="font-family: MathJax_Math-italic;">z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.342em; left: 0.539em;"><span class="mi" id="MathJax-Span-82" style="font-size: 70.7%; font-family: MathJax_Math-italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mi" id="MathJax-Span-83" style="font-family: MathJax_Math-italic;">A</span><span class="mi" id="MathJax-Span-84" style="font-family: MathJax_Math-italic;">z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-85" style="font-family: MathJax_Main; padding-left: 0.301em;">&gt;</span><span class="mn" id="MathJax-Span-86" style="font-family: MathJax_Main; padding-left: 0.301em;">0</span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.139em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mi>T</mi></msup><mi>A</mi><mi>z</mi><mo>&gt;</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-11">z^T A z > 0</script>，则称矩阵 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-12-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-87" style="width: 0.955em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.777em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1000.78em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-88"><span class="mi" id="MathJax-Span-89" style="font-family: MathJax_Math-italic;">A</span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-12">A</script> 是正定的。正定矩阵的行列式必然大于 0， 所有特征值也必然 &gt; 0。相对应的，半正定矩阵的行列式必然 ≥ 0。<a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/[%3E%3E%20%E6%AD%A3%E4%BA%A4%E7%9F%A9%E9%98%B5%E7%9F%A5%E8%AF%86%E6%89%A9%E5%B1%95](https://zh.wikipedia.org/wiki/%E6%AD%A3%E4%BA%A4%E7%9F%A9%E9%98%B5">&gt;&gt; 正定矩阵知识扩展</a></li>
</ul>
<blockquote>
<p>转置与共轭转置</p>
</blockquote>
<p class=" has-jax has-jax has-jax has-jax has-jax has-jax has-jax">矩阵的转置（transpose）是最简单的一种矩阵变换。简单来说，若 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-13-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-90" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.967em, 1002.74em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-91"><span class="mi" id="MathJax-Span-92" style="font-family: MathJax_Math-italic;">m</span><span class="mo" id="MathJax-Span-93" style="font-family: MathJax_Main; padding-left: 0.241em;">×</span><span class="mi" id="MathJax-Span-94" style="font-family: MathJax_Math-italic; padding-left: 0.241em;">n</span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 0.718em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>×</mo><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-13">m\times n</script> 的矩阵 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-14-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-95" style="width: 1.313em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.074em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1001.01em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-96"><span class="texatom" id="MathJax-Span-97"><span class="mrow" id="MathJax-Span-98"><span class="mi" id="MathJax-Span-99" style="font-family: MathJax_Main-bold;">M</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-14">\mathbf M</script> 的转置记为 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-15-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-100" style="width: 2.027em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1001.67em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-101"><span class="msubsup" id="MathJax-Span-102"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-103"><span class="mrow" id="MathJax-Span-104"><span class="mi" id="MathJax-Span-105" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-106"><span class="mrow" id="MathJax-Span-107"><span class="texatom" id="MathJax-Span-108"><span class="mrow" id="MathJax-Span-109"><span class="mi" id="MathJax-Span-110" style="font-size: 70.7%; font-family: MathJax_SansSerif;">T</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">T</mi></mrow></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-15">\mathbf M^{\mathsf T}</script>；则 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-16-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-111" style="width: 2.027em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1001.67em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-112"><span class="msubsup" id="MathJax-Span-113"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-114"><span class="mrow" id="MathJax-Span-115"><span class="mi" id="MathJax-Span-116" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-117"><span class="mrow" id="MathJax-Span-118"><span class="texatom" id="MathJax-Span-119"><span class="mrow" id="MathJax-Span-120"><span class="mi" id="MathJax-Span-121" style="font-size: 70.7%; font-family: MathJax_SansSerif;">T</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">T</mi></mrow></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-16">\mathbf M^{\mathsf T}</script> 是一个 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-17-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-122" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.967em, 1002.74em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-123"><span class="mi" id="MathJax-Span-124" style="font-family: MathJax_Math-italic;">n</span><span class="mo" id="MathJax-Span-125" style="font-family: MathJax_Main; padding-left: 0.241em;">×</span><span class="mi" id="MathJax-Span-126" style="font-family: MathJax_Math-italic; padding-left: 0.241em;">m</span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 0.718em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>×</mo><mi>m</mi></math></span></span><script type="math/tex" id="MathJax-Element-17">n\times m</script> 的矩阵，并且 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-18-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-127" style="width: 6.253em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.182em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1005.18em, 3.217em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-128"><span class="msubsup" id="MathJax-Span-129"><span style="display: inline-block; position: relative; width: 1.908em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-130"><span class="mrow" id="MathJax-Span-131"><span class="mi" id="MathJax-Span-132" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.807em; left: 1.074em;"><span class="texatom" id="MathJax-Span-133"><span class="mrow" id="MathJax-Span-134"><span class="mi" id="MathJax-Span-135" style="font-size: 70.7%; font-family: MathJax_Math-italic;">i</span><span class="mo" id="MathJax-Span-136" style="font-size: 70.7%; font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-137" style="font-size: 70.7%; font-family: MathJax_Math-italic;">j</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-138" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="msubsup" id="MathJax-Span-139" style="padding-left: 0.301em;"><span style="display: inline-block; position: relative; width: 1.908em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-140"><span class="mrow" id="MathJax-Span-141"><span class="mi" id="MathJax-Span-142" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.336em, 1000.54em, 4.17em, -999.997em); top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-143"><span class="mrow" id="MathJax-Span-144"><span class="texatom" id="MathJax-Span-145"><span class="mrow" id="MathJax-Span-146"><span class="mi" id="MathJax-Span-147" style="font-size: 70.7%; font-family: MathJax_SansSerif;">T</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.336em, 1000.84em, 4.289em, -999.997em); top: -3.747em; left: 1.074em;"><span class="texatom" id="MathJax-Span-148"><span class="mrow" id="MathJax-Span-149"><span class="mi" id="MathJax-Span-150" style="font-size: 70.7%; font-family: MathJax_Math-italic;">j</span><span class="mo" id="MathJax-Span-151" style="font-size: 70.7%; font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-152" style="font-size: 70.7%; font-family: MathJax_Math-italic;">i</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.568em; border-left: 0px solid; width: 0px; height: 1.718em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><msubsup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>,</mo><mi>i</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">T</mi></mrow></mrow></msubsup></math></span></span><script type="math/tex" id="MathJax-Element-18">\mathbf M_{i,j} = \mathbf M^{\mathsf T}_{j,i}</script>。因此，矩阵的转置相当于将矩阵按照主对角线翻转；同时，我们不难得出 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-19-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-153" style="width: 6.729em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.598em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.372em, 1005.6em, 3.158em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-154"><span class="texatom" id="MathJax-Span-155"><span class="mrow" id="MathJax-Span-156"><span class="mi" id="MathJax-Span-157" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="mo" id="MathJax-Span-158" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="texatom" id="MathJax-Span-159" style="padding-left: 0.301em;"><span class="mrow" id="MathJax-Span-160"><span class="mo" id="MathJax-Span-161" style="vertical-align: 0em;"><span style="font-family: MathJax_Size1;">(</span></span></span></span><span class="msubsup" id="MathJax-Span-162"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-163"><span class="mrow" id="MathJax-Span-164"><span class="mi" id="MathJax-Span-165" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-166"><span class="mrow" id="MathJax-Span-167"><span class="texatom" id="MathJax-Span-168"><span class="mrow" id="MathJax-Span-169"><span class="mi" id="MathJax-Span-170" style="font-size: 70.7%; font-family: MathJax_SansSerif;">T</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-171" style=""><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(2.979em, 1000.3em, 4.527em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-172" style=""><span class="mrow" id="MathJax-Span-173"><span class="mo" id="MathJax-Span-174" style="vertical-align: 0em;"><span style="font-family: MathJax_Size1;">)</span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.58em; left: 0.479em;"><span class="texatom" id="MathJax-Span-175"><span class="mrow" id="MathJax-Span-176"><span class="texatom" id="MathJax-Span-177"><span class="mrow" id="MathJax-Span-178"><span class="mi" id="MathJax-Span-179" style="font-size: 70.7%; font-family: MathJax_SansSerif;">T</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.496em; border-left: 0px solid; width: 0px; height: 1.861em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mo>=</mo><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">T</mi></mrow></mrow></msup><msup><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">T</mi></mrow></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-19">\mathbf M = \bigl(\mathbf M^{\mathsf T}\bigr)^{\mathsf T}</script>。</p>
<p><img src="https://i.imgur.com/fiYG6r9.gif" alt="" data-bd-imgshare-binded="1"></p>
<p class=" has-jax has-jax has-jax">矩阵的共轭转置（conjugate transpose）可能是倒数第二简单的矩阵变换。共轭转置只需要在转置的基础上，再叠加复数的共轭即可。因此，若以 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-20-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-180" style="width: 2.027em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1001.67em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-181"><span class="msubsup" id="MathJax-Span-182"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-183"><span class="mrow" id="MathJax-Span-184"><span class="mi" id="MathJax-Span-185" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-186"><span class="mrow" id="MathJax-Span-187"><span class="texatom" id="MathJax-Span-188"><span class="mrow" id="MathJax-Span-189"><span class="mi" id="MathJax-Span-190" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-20">\mathbf M^{\mathsf H}</script> 记矩阵 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-21-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-191" style="width: 1.313em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.074em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1001.01em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-192"><span class="texatom" id="MathJax-Span-193"><span class="mrow" id="MathJax-Span-194"><span class="mi" id="MathJax-Span-195" style="font-family: MathJax_Main-bold;">M</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-21">\mathbf M</script> 的共轭转置，则有 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-22-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msub&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-196" style="width: 8.039em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.67em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.193em, 1006.67em, 3.336em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-197"><span class="msubsup" id="MathJax-Span-198"><span style="display: inline-block; position: relative; width: 1.908em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-199"><span class="mrow" id="MathJax-Span-200"><span class="mi" id="MathJax-Span-201" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.807em; left: 1.074em;"><span class="texatom" id="MathJax-Span-202"><span class="mrow" id="MathJax-Span-203"><span class="mi" id="MathJax-Span-204" style="font-size: 70.7%; font-family: MathJax_Math-italic;">i</span><span class="mo" id="MathJax-Span-205" style="font-size: 70.7%; font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-206" style="font-size: 70.7%; font-family: MathJax_Math-italic;">j</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-207" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="munderover" id="MathJax-Span-208" style="padding-left: 0.301em;"><span style="display: inline-block; position: relative; width: 3.396em; height: 0px;"><span style="position: absolute; clip: rect(2.92em, 1003.4em, 4.705em, -999.997em); top: -3.985em; left: 0.003em;"><span class="mrow" id="MathJax-Span-209"><span class="texatom" id="MathJax-Span-210" style=""><span class="mrow" id="MathJax-Span-211"><span class="mo" id="MathJax-Span-212" style="vertical-align: 0em;"><span style="font-family: MathJax_Size1;">(</span></span></span></span><span class="msubsup" id="MathJax-Span-213"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-214"><span class="mrow" id="MathJax-Span-215"><span class="mi" id="MathJax-Span-216" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-217"><span class="mrow" id="MathJax-Span-218"><span class="texatom" id="MathJax-Span-219"><span class="mrow" id="MathJax-Span-220"><span class="mi" id="MathJax-Span-221" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-222" style=""><span style="display: inline-block; position: relative; width: 1.253em; height: 0px;"><span style="position: absolute; clip: rect(2.979em, 1000.3em, 4.527em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-223" style=""><span class="mrow" id="MathJax-Span-224"><span class="mo" id="MathJax-Span-225" style="vertical-align: 0em;"><span style="font-family: MathJax_Size1;">)</span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.628em; left: 0.479em;"><span class="texatom" id="MathJax-Span-226"><span class="mrow" id="MathJax-Span-227"><span class="mi" id="MathJax-Span-228" style="font-size: 70.7%; font-family: MathJax_Math-italic;">j</span><span class="mo" id="MathJax-Span-229" style="font-size: 70.7%; font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-230" style="font-size: 70.7%; font-family: MathJax_Math-italic;">i</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.277em, 1003.4em, 3.812em, -999.997em); top: -4.699em; left: 0em;"><span class="mo" id="MathJax-Span-231" style=""><span style="display: inline-block; position: relative; width: 3.396em; height: 0px;"><span style="position: absolute; top: -3.985em; left: -0.057em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.985em; left: 3.098em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.985em; left: 0.122em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.985em; left: 0.301em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.985em; left: 0.539em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.985em; left: 0.717em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.985em; left: 0.896em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.985em; left: 1.134em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.985em; left: 1.313em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.985em; left: 1.551em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.985em; left: 1.729em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.985em; left: 1.908em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.985em; left: 2.146em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.985em; left: 2.324em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.985em; left: 2.503em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.985em; left: 2.741em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.985em; left: 2.92em;"><span style="font-size: 70.7%; font-family: MathJax_Main;">¯</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.711em; border-left: 0px solid; width: 0px; height: 2.289em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><mover><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><msub><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>,</mo><mi>i</mi></mrow></msub></mrow><mo accent="false">¯</mo></mover></math></span></span><script type="math/tex" id="MathJax-Element-22">\mathbf M_{i,j} = \overline{\bigl(\mathbf M^{\mathsf H}\bigr)_{j,i}}</script>。</p>
<blockquote>
<p>酉矩阵</p>
</blockquote>
<p class=" has-jax has-jax has-jax has-jax">酉矩阵（unitary matrix）是一种特殊的方阵，它满足<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-23-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-232" style="width: 10.539em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.753em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1008.69em, 2.979em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-233"><span class="texatom" id="MathJax-Span-234"><span class="mrow" id="MathJax-Span-235"><span class="mi" id="MathJax-Span-236" style="font-family: MathJax_Main-bold;">U</span></span></span><span class="msubsup" id="MathJax-Span-237"><span style="display: inline-block; position: relative; width: 1.491em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.84em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-238"><span class="mrow" id="MathJax-Span-239"><span class="mi" id="MathJax-Span-240" style="font-family: MathJax_Main-bold;">U</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.896em;"><span class="texatom" id="MathJax-Span-241"><span class="mrow" id="MathJax-Span-242"><span class="texatom" id="MathJax-Span-243"><span class="mrow" id="MathJax-Span-244"><span class="mi" id="MathJax-Span-245" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-246" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="msubsup" id="MathJax-Span-247" style="padding-left: 0.301em;"><span style="display: inline-block; position: relative; width: 1.491em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.84em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-248"><span class="mrow" id="MathJax-Span-249"><span class="mi" id="MathJax-Span-250" style="font-family: MathJax_Main-bold;">U</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.896em;"><span class="texatom" id="MathJax-Span-251"><span class="mrow" id="MathJax-Span-252"><span class="texatom" id="MathJax-Span-253"><span class="mrow" id="MathJax-Span-254"><span class="mi" id="MathJax-Span-255" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="texatom" id="MathJax-Span-256"><span class="mrow" id="MathJax-Span-257"><span class="mi" id="MathJax-Span-258" style="font-family: MathJax_Main-bold;">U</span></span></span><span class="mo" id="MathJax-Span-259" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="msubsup" id="MathJax-Span-260" style="padding-left: 0.301em;"><span style="display: inline-block; position: relative; width: 0.955em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-261" style="font-family: MathJax_Math-italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.807em; left: 0.42em;"><span class="mi" id="MathJax-Span-262" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-263" style="font-family: MathJax_Main;">.</span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.432em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mo>=</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow><mo>=</mo><msub><mi>I</mi><mi>n</mi></msub><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-23">\mathbf U\mathbf U^{\mathsf H} = \mathbf U^{\mathsf H}\mathbf U = I_n.</script> 酉矩阵实际上是推广的正交矩阵（orthogonal matrix）；当酉矩阵中的元素均为实数时，酉矩阵实际就是正交矩阵。另一方面，由于 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-24-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-264" style="width: 12.086em; display: inline-block;"><span style="display: inline-block; position: relative; width: 10.063em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1010.06em, 2.979em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-265"><span class="texatom" id="MathJax-Span-266"><span class="mrow" id="MathJax-Span-267"><span class="mi" id="MathJax-Span-268" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="msubsup" id="MathJax-Span-269"><span style="display: inline-block; position: relative; width: 2.086em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-270"><span class="mrow" id="MathJax-Span-271"><span class="mi" id="MathJax-Span-272" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-273"><span class="mrow" id="MathJax-Span-274"><span class="mo" id="MathJax-Span-275" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-276" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-277" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="msubsup" id="MathJax-Span-278" style="padding-left: 0.301em;"><span style="display: inline-block; position: relative; width: 2.086em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-279"><span class="mrow" id="MathJax-Span-280"><span class="mi" id="MathJax-Span-281" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-282"><span class="mrow" id="MathJax-Span-283"><span class="mo" id="MathJax-Span-284" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-285" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="texatom" id="MathJax-Span-286"><span class="mrow" id="MathJax-Span-287"><span class="mi" id="MathJax-Span-288" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="mo" id="MathJax-Span-289" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="msubsup" id="MathJax-Span-290" style="padding-left: 0.301em;"><span style="display: inline-block; position: relative; width: 0.955em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-291" style="font-family: MathJax_Math-italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.807em; left: 0.42em;"><span class="mi" id="MathJax-Span-292" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.432em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mo>=</mo><msub><mi>I</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-24">\mathbf M\mathbf M^{-1} = \mathbf M^{-1}\mathbf M = I_n</script>，所以酉矩阵 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-25-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-293" style="width: 1.074em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.896em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1000.84em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-294"><span class="texatom" id="MathJax-Span-295"><span class="mrow" id="MathJax-Span-296"><span class="mi" id="MathJax-Span-297" style="font-family: MathJax_Main-bold;">U</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-25">\mathbf U</script> 满足 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-26-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-298" style="width: 5.658em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.705em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1004.71em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-299"><span class="msubsup" id="MathJax-Span-300"><span style="display: inline-block; position: relative; width: 1.848em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.84em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-301"><span class="mrow" id="MathJax-Span-302"><span class="mi" id="MathJax-Span-303" style="font-family: MathJax_Main-bold;">U</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.896em;"><span class="texatom" id="MathJax-Span-304"><span class="mrow" id="MathJax-Span-305"><span class="mo" id="MathJax-Span-306" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-307" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-308" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="msubsup" id="MathJax-Span-309" style="padding-left: 0.301em;"><span style="display: inline-block; position: relative; width: 1.491em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.84em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-310"><span class="mrow" id="MathJax-Span-311"><span class="mi" id="MathJax-Span-312" style="font-family: MathJax_Main-bold;">U</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.896em;"><span class="texatom" id="MathJax-Span-313"><span class="mrow" id="MathJax-Span-314"><span class="texatom" id="MathJax-Span-315"><span class="mrow" id="MathJax-Span-316"><span class="mi" id="MathJax-Span-317" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-26">\mathbf U^{-1} = \mathbf U^{\mathsf H}</script>；事实上，这是一个矩阵是酉矩阵的充分必要条件。</p>
<blockquote>
<p>正规矩阵</p>
</blockquote>
<p class=" has-jax has-jax has-jax has-jax">同酉矩阵一样，正规矩阵（normal matrix）也是一种特殊的方阵，它要求在矩阵乘法的意义下与它的共轭转置矩阵满足交换律。这也就是说，若矩阵 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-27-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-318" style="width: 1.313em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.074em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1001.01em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-319"><span class="texatom" id="MathJax-Span-320"><span class="mrow" id="MathJax-Span-321"><span class="mi" id="MathJax-Span-322" style="font-family: MathJax_Main-bold;">M</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-27">\mathbf M</script> 满足如下条件，则称其为正规矩阵：<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-28-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-323" style="width: 8.693em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.205em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1007.15em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-324"><span class="texatom" id="MathJax-Span-325"><span class="mrow" id="MathJax-Span-326"><span class="mi" id="MathJax-Span-327" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="msubsup" id="MathJax-Span-328"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-329"><span class="mrow" id="MathJax-Span-330"><span class="mi" id="MathJax-Span-331" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-332"><span class="mrow" id="MathJax-Span-333"><span class="texatom" id="MathJax-Span-334"><span class="mrow" id="MathJax-Span-335"><span class="mi" id="MathJax-Span-336" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-337" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="msubsup" id="MathJax-Span-338" style="padding-left: 0.301em;"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-339"><span class="mrow" id="MathJax-Span-340"><span class="mi" id="MathJax-Span-341" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-342"><span class="mrow" id="MathJax-Span-343"><span class="texatom" id="MathJax-Span-344"><span class="mrow" id="MathJax-Span-345"><span class="mi" id="MathJax-Span-346" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="texatom" id="MathJax-Span-347"><span class="mrow" id="MathJax-Span-348"><span class="mi" id="MathJax-Span-349" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="mo" id="MathJax-Span-350" style="font-family: MathJax_Main;">.</span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mo>=</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-28">\mathbf M\mathbf M^{\mathsf H} = \mathbf M^{\mathsf H}\mathbf M.</script>。显而易见，复系数的酉矩阵和实系数的正交矩阵都是正规矩阵。显而易见，正规矩阵并不只有酉矩阵或正交矩阵。例如说，矩阵 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-29-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-351" style="width: 9.824em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.158em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(2.443em, 1007.86em, 6.61em, -999.997em); top: -4.759em; left: 0em;"><span class="mrow" id="MathJax-Span-352"><span class="texatom" id="MathJax-Span-353"><span class="mrow" id="MathJax-Span-354"><span class="mi" id="MathJax-Span-355" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="mo" id="MathJax-Span-356" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="mrow" id="MathJax-Span-357" style="padding-left: 0.301em;"><span class="mo" id="MathJax-Span-358" style="vertical-align: 2.146em;"><span style="display: inline-block; position: relative; width: 0.896em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎛<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: -0.83em; left: 0em;">⎝<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.783em; left: 0em;">⎜<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mtable" id="MathJax-Span-359" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 3.515em; height: 0px;"><span style="position: absolute; clip: rect(2.265em, 1000.48em, 6.074em, -999.997em); top: -4.461em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -5.354em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-360"><span class="mrow" id="MathJax-Span-361"><span class="mn" id="MathJax-Span-362" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.926em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-369"><span class="mrow" id="MathJax-Span-370"><span class="mn" id="MathJax-Span-371" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -2.557em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-378"><span class="mrow" id="MathJax-Span-379"><span class="mn" id="MathJax-Span-380" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.467em;"></span></span><span style="position: absolute; clip: rect(2.265em, 1000.48em, 6.134em, -999.997em); top: -4.461em; left: 1.491em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -5.354em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-363"><span class="mrow" id="MathJax-Span-364"><span class="mn" id="MathJax-Span-365" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -3.926em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-372"><span class="mrow" id="MathJax-Span-373"><span class="mn" id="MathJax-Span-374" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -2.557em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-381"><span class="mrow" id="MathJax-Span-382"><span class="mn" id="MathJax-Span-383" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.467em;"></span></span><span style="position: absolute; clip: rect(2.265em, 1000.48em, 6.074em, -999.997em); top: -4.461em; left: 2.979em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -5.354em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-366"><span class="mrow" id="MathJax-Span-367"><span class="mn" id="MathJax-Span-368" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -3.926em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-375"><span class="mrow" id="MathJax-Span-376"><span class="mn" id="MathJax-Span-377" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -2.557em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-384"><span class="mrow" id="MathJax-Span-385"><span class="mn" id="MathJax-Span-386" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.467em;"></span></span></span></span><span class="mo" id="MathJax-Span-387" style="vertical-align: 2.146em;"><span style="display: inline-block; position: relative; width: 0.896em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎞<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: -0.83em; left: 0em;">⎠<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.783em; left: 0em;">⎟<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.765em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.068em; border-left: 0px solid; width: 0px; height: 4.718em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mo>=</mo><mrow><mo>(</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span></span><script type="math/tex" id="MathJax-Element-29">\mathbf M = \begin{pmatrix}1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1\end{pmatrix}</script> 即是一个正规矩阵，但它显然不是酉矩阵或正交矩阵；因为<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-30-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-388" style="width: 17.086em; display: inline-block;"><span style="display: inline-block; position: relative; width: 14.229em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(2.443em, 1014.17em, 6.61em, -999.997em); top: -4.759em; left: 0em;"><span class="mrow" id="MathJax-Span-389"><span class="texatom" id="MathJax-Span-390"><span class="mrow" id="MathJax-Span-391"><span class="mi" id="MathJax-Span-392" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="msubsup" id="MathJax-Span-393"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-394"><span class="mrow" id="MathJax-Span-395"><span class="mi" id="MathJax-Span-396" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-397"><span class="mrow" id="MathJax-Span-398"><span class="texatom" id="MathJax-Span-399"><span class="mrow" id="MathJax-Span-400"><span class="mi" id="MathJax-Span-401" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-402" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="mrow" id="MathJax-Span-403" style="padding-left: 0.301em;"><span class="mo" id="MathJax-Span-404" style="vertical-align: 2.146em;"><span style="display: inline-block; position: relative; width: 0.896em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎛<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: -0.83em; left: 0em;">⎝<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.783em; left: 0em;">⎜<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mtable" id="MathJax-Span-405" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 3.515em; height: 0px;"><span style="position: absolute; clip: rect(2.265em, 1000.48em, 6.074em, -999.997em); top: -4.461em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -5.354em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-406"><span class="mrow" id="MathJax-Span-407"><span class="mn" id="MathJax-Span-408" style="font-family: MathJax_Main;">2</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -3.926em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-415"><span class="mrow" id="MathJax-Span-416"><span class="mn" id="MathJax-Span-417" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -2.557em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-424"><span class="mrow" id="MathJax-Span-425"><span class="mn" id="MathJax-Span-426" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.467em;"></span></span><span style="position: absolute; clip: rect(2.265em, 1000.48em, 6.074em, -999.997em); top: -4.461em; left: 1.491em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -5.354em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-409"><span class="mrow" id="MathJax-Span-410"><span class="mn" id="MathJax-Span-411" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.926em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-418"><span class="mrow" id="MathJax-Span-419"><span class="mn" id="MathJax-Span-420" style="font-family: MathJax_Main;">2</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -2.557em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-427"><span class="mrow" id="MathJax-Span-428"><span class="mn" id="MathJax-Span-429" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.467em;"></span></span><span style="position: absolute; clip: rect(2.265em, 1000.48em, 6.074em, -999.997em); top: -4.461em; left: 2.979em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -5.354em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-412"><span class="mrow" id="MathJax-Span-413"><span class="mn" id="MathJax-Span-414" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -3.926em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-421"><span class="mrow" id="MathJax-Span-422"><span class="mn" id="MathJax-Span-423" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -2.557em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-430"><span class="mrow" id="MathJax-Span-431"><span class="mn" id="MathJax-Span-432" style="font-family: MathJax_Main;">2</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.467em;"></span></span></span></span><span class="mo" id="MathJax-Span-433" style="vertical-align: 2.146em;"><span style="display: inline-block; position: relative; width: 0.896em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎞<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: -0.83em; left: 0em;">⎠<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.783em; left: 0em;">⎟<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span class="mo" id="MathJax-Span-434" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="msubsup" id="MathJax-Span-435" style="padding-left: 0.301em;"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-436"><span class="mrow" id="MathJax-Span-437"><span class="mi" id="MathJax-Span-438" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-439"><span class="mrow" id="MathJax-Span-440"><span class="texatom" id="MathJax-Span-441"><span class="mrow" id="MathJax-Span-442"><span class="mi" id="MathJax-Span-443" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="texatom" id="MathJax-Span-444"><span class="mrow" id="MathJax-Span-445"><span class="mi" id="MathJax-Span-446" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="mo" id="MathJax-Span-447" style="font-family: MathJax_Main;">.</span></span><span style="display: inline-block; width: 0px; height: 4.765em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.068em; border-left: 0px solid; width: 0px; height: 4.718em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mo>=</mo><mrow><mo>(</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>=</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-30">\mathbf M\mathbf M^{\mathsf H} = \begin{pmatrix}2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2\end{pmatrix} = \mathbf M^{\mathsf H}\mathbf M.</script></p>
<blockquote>
<p>谱定理和谱分解</p>
</blockquote>
<p class=" has-jax has-jax has-jax has-jax">矩阵的对角化是线性代数中的一个重要命题。谱定理（spectral theorem）给出了方阵对角化的一个结论：若矩阵 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-31-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-448" style="width: 1.313em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.074em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1001.01em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-449"><span class="texatom" id="MathJax-Span-450"><span class="mrow" id="MathJax-Span-451"><span class="mi" id="MathJax-Span-452" style="font-family: MathJax_Main-bold;">M</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-31">\mathbf M</script> 是一个正规矩阵，则存在酉矩阵 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-32-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-453" style="width: 1.074em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.896em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1000.84em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-454"><span class="texatom" id="MathJax-Span-455"><span class="mrow" id="MathJax-Span-456"><span class="mi" id="MathJax-Span-457" style="font-family: MathJax_Main-bold;">U</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-32">\mathbf U</script>，以及对角矩阵 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-33-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;&amp;#x039B;&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-458" style="width: 1.015em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.836em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1000.78em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-459"><span class="texatom" id="MathJax-Span-460"><span class="mrow" id="MathJax-Span-461"><span class="mi" id="MathJax-Span-462" style="font-family: MathJax_Main-bold;">Λ</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">Λ</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-33">\mathbf \Lambda</script>，使得<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-34-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;&amp;#x039B;&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-463" style="width: 7.146em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.955em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1005.9em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-464"><span class="texatom" id="MathJax-Span-465"><span class="mrow" id="MathJax-Span-466"><span class="mi" id="MathJax-Span-467" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="mo" id="MathJax-Span-468" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="texatom" id="MathJax-Span-469" style="padding-left: 0.301em;"><span class="mrow" id="MathJax-Span-470"><span class="mi" id="MathJax-Span-471" style="font-family: MathJax_Main-bold;">U</span></span></span><span class="texatom" id="MathJax-Span-472"><span class="mrow" id="MathJax-Span-473"><span class="mi" id="MathJax-Span-474" style="font-family: MathJax_Main-bold;">Λ</span></span></span><span class="msubsup" id="MathJax-Span-475"><span style="display: inline-block; position: relative; width: 1.491em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.84em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-476"><span class="mrow" id="MathJax-Span-477"><span class="mi" id="MathJax-Span-478" style="font-family: MathJax_Main-bold;">U</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.896em;"><span class="texatom" id="MathJax-Span-479"><span class="mrow" id="MathJax-Span-480"><span class="texatom" id="MathJax-Span-481"><span class="mrow" id="MathJax-Span-482"><span class="mi" id="MathJax-Span-483" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-484" style="font-family: MathJax_Main;">.</span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">Λ</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-34">\mathbf M = \mathbf U\mathbf \Lambda\mathbf U^{\mathsf H}.</script><br>这也就是说，正规矩阵，可经由酉变换，分解为对角矩阵；这种矩阵分解的方式，称为谱分解（spectral decomposition）。</p>
<hr>
<h1 id="SVD-的计算方法"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#SVD-%E7%9A%84%E8%AE%A1%E7%AE%97%E6%96%B9%E6%B3%95" class="headerlink" title="SVD 的计算方法"></a>SVD 的计算方法</h1><h2 id="SVD-与特征值"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#SVD-%E4%B8%8E%E7%89%B9%E5%BE%81%E5%80%BC" class="headerlink" title="SVD 与特征值"></a>SVD 与特征值</h2><p class=" has-jax has-jax">现在，假设矩阵 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-35-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-485" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1002.74em, 2.979em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-486"><span class="msubsup" id="MathJax-Span-487"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-488"><span class="mrow" id="MathJax-Span-489"><span class="mi" id="MathJax-Span-490" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.807em; left: 1.074em;"><span class="texatom" id="MathJax-Span-491"><span class="mrow" id="MathJax-Span-492"><span class="mi" id="MathJax-Span-493" style="font-size: 70.7%; font-family: MathJax_Math-italic;">m</span><span class="mo" id="MathJax-Span-494" style="font-size: 70.7%; font-family: MathJax_Main;">×</span><span class="mi" id="MathJax-Span-495" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.146em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-35">\mathbf M_{m\times n}</script> 的 SVD 分解是<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-36-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;&amp;#x03A3;&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-496" style="width: 7.086em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.896em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1005.84em, 2.979em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-497"><span class="texatom" id="MathJax-Span-498"><span class="mrow" id="MathJax-Span-499"><span class="mi" id="MathJax-Span-500" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="mo" id="MathJax-Span-501" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="texatom" id="MathJax-Span-502" style="padding-left: 0.301em;"><span class="mrow" id="MathJax-Span-503"><span class="mi" id="MathJax-Span-504" style="font-family: MathJax_Main-bold;">U</span></span></span><span class="texatom" id="MathJax-Span-505"><span class="mrow" id="MathJax-Span-506"><span class="mi" id="MathJax-Span-507" style="font-family: MathJax_Main-bold;">Σ</span></span></span><span class="msubsup" id="MathJax-Span-508"><span style="display: inline-block; position: relative; width: 1.432em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.84em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-509"><span class="mrow" id="MathJax-Span-510"><span class="mi" id="MathJax-Span-511" style="font-family: MathJax_Main-bold;">V</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.896em;"><span class="texatom" id="MathJax-Span-512"><span class="mrow" id="MathJax-Span-513"><span class="texatom" id="MathJax-Span-514"><span class="mrow" id="MathJax-Span-515"><span class="mi" id="MathJax-Span-516" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-517" style="font-family: MathJax_Main;">;</span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.432em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">Σ</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">V</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mo>;</mo></math></span></span><script type="math/tex" id="MathJax-Element-36">\mathbf M = \mathbf U\mathbf\Sigma\mathbf V^{\mathsf H};</script>那么，我们有</p>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-37-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;right left right left right left right left right left right left&quot; rowspacing=&quot;3pt&quot; columnspacing=&quot;0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em&quot; displaystyle=&quot;true&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;&amp;#x03A3;&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;&amp;#x03A3;&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;&amp;#x03A3;&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;&amp;#x03A3;&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;&amp;#x03A3;&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;&amp;#x03A3;&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;&amp;#x03A3;&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;&amp;#x03A3;&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-518" style="width: 21.789em; display: inline-block;"><span style="display: inline-block; position: relative; width: 18.158em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(0.896em, 1017.98em, 3.872em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-519"><span class="mtable" id="MathJax-Span-520" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 17.801em; height: 0px;"><span style="position: absolute; clip: rect(2.265em, 1002.74em, 5.003em, -999.997em); top: -3.985em; left: 0em;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px;"><span style="position: absolute; clip: rect(2.92em, 1002.74em, 4.17em, -999.997em); top: -4.64em; right: 0em;"><span class="mtd" id="MathJax-Span-521"><span class="mrow" id="MathJax-Span-522"><span class="texatom" id="MathJax-Span-523"><span class="mrow" id="MathJax-Span-524"><span class="mi" id="MathJax-Span-525" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="msubsup" id="MathJax-Span-526"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-527"><span class="mrow" id="MathJax-Span-528"><span class="mi" id="MathJax-Span-529" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-530"><span class="mrow" id="MathJax-Span-531"><span class="texatom" id="MathJax-Span-532"><span class="mrow" id="MathJax-Span-533"><span class="mi" id="MathJax-Span-534" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(2.92em, 1002.74em, 4.17em, -999.997em); top: -3.152em; right: 0em;"><span class="mtd" id="MathJax-Span-604"><span class="mrow" id="MathJax-Span-605"><span class="msubsup" id="MathJax-Span-606"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-607"><span class="mrow" id="MathJax-Span-608"><span class="mi" id="MathJax-Span-609" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-610"><span class="mrow" id="MathJax-Span-611"><span class="texatom" id="MathJax-Span-612"><span class="mrow" id="MathJax-Span-613"><span class="mi" id="MathJax-Span-614" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="texatom" id="MathJax-Span-615"><span class="mrow" id="MathJax-Span-616"><span class="mi" id="MathJax-Span-617" style="font-family: MathJax_Main-bold;">M</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(2.265em, 1015em, 5.241em, -999.997em); top: -3.985em; left: 2.741em;"><span style="display: inline-block; position: relative; width: 15.003em; height: 0px;"><span style="position: absolute; clip: rect(2.92em, 1015em, 4.408em, -999.997em); top: -4.64em; left: 0em;"><span class="mtd" id="MathJax-Span-535"><span class="mrow" id="MathJax-Span-536"><span class="mi" id="MathJax-Span-537"></span><span class="texatom" id="MathJax-Span-538"><span class="mrow" id="MathJax-Span-539"></span></span><span class="mo" id="MathJax-Span-540" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="texatom" id="MathJax-Span-541" style="padding-left: 0.301em;"><span class="mrow" id="MathJax-Span-542"><span class="mi" id="MathJax-Span-543" style="font-family: MathJax_Main-bold;">U</span></span></span><span class="texatom" id="MathJax-Span-544"><span class="mrow" id="MathJax-Span-545"><span class="mi" id="MathJax-Span-546" style="font-family: MathJax_Main-bold;">Σ</span></span></span><span class="msubsup" id="MathJax-Span-547"><span style="display: inline-block; position: relative; width: 1.432em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.84em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-548"><span class="mrow" id="MathJax-Span-549"><span class="mi" id="MathJax-Span-550" style="font-family: MathJax_Main-bold;">V</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.896em;"><span class="texatom" id="MathJax-Span-551"><span class="mrow" id="MathJax-Span-552"><span class="texatom" id="MathJax-Span-553"><span class="mrow" id="MathJax-Span-554"><span class="mi" id="MathJax-Span-555" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="texatom" id="MathJax-Span-556"><span class="mrow" id="MathJax-Span-557"><span class="mi" id="MathJax-Span-558" style="font-family: MathJax_Main-bold;">V</span></span></span><span class="msubsup" id="MathJax-Span-559"><span style="display: inline-block; position: relative; width: 1.432em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.78em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-560"><span class="mrow" id="MathJax-Span-561"><span class="mi" id="MathJax-Span-562" style="font-family: MathJax_Main-bold;">Σ</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.836em;"><span class="texatom" id="MathJax-Span-563"><span class="mrow" id="MathJax-Span-564"><span class="texatom" id="MathJax-Span-565"><span class="mrow" id="MathJax-Span-566"><span class="mi" id="MathJax-Span-567" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-568"><span style="display: inline-block; position: relative; width: 1.491em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.84em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-569"><span class="mrow" id="MathJax-Span-570"><span class="mi" id="MathJax-Span-571" style="font-family: MathJax_Main-bold;">U</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.896em;"><span class="texatom" id="MathJax-Span-572"><span class="mrow" id="MathJax-Span-573"><span class="texatom" id="MathJax-Span-574"><span class="mrow" id="MathJax-Span-575"><span class="mi" id="MathJax-Span-576" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-577" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="texatom" id="MathJax-Span-578" style="padding-left: 0.301em;"><span class="mrow" id="MathJax-Span-579"><span class="mi" id="MathJax-Span-580" style="font-family: MathJax_Main-bold;">U</span></span></span><span class="mo" id="MathJax-Span-581" style="font-family: MathJax_Main;">(</span><span class="texatom" id="MathJax-Span-582"><span class="mrow" id="MathJax-Span-583"><span class="mi" id="MathJax-Span-584" style="font-family: MathJax_Main-bold;">Σ</span></span></span><span class="msubsup" id="MathJax-Span-585"><span style="display: inline-block; position: relative; width: 1.432em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.78em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-586"><span class="mrow" id="MathJax-Span-587"><span class="mi" id="MathJax-Span-588" style="font-family: MathJax_Main-bold;">Σ</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.836em;"><span class="texatom" id="MathJax-Span-589"><span class="mrow" id="MathJax-Span-590"><span class="texatom" id="MathJax-Span-591"><span class="mrow" id="MathJax-Span-592"><span class="mi" id="MathJax-Span-593" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-594" style="font-family: MathJax_Main;">)</span><span class="msubsup" id="MathJax-Span-595"><span style="display: inline-block; position: relative; width: 1.491em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.84em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-596"><span class="mrow" id="MathJax-Span-597"><span class="mi" id="MathJax-Span-598" style="font-family: MathJax_Main-bold;">U</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.896em;"><span class="texatom" id="MathJax-Span-599"><span class="mrow" id="MathJax-Span-600"><span class="texatom" id="MathJax-Span-601"><span class="mrow" id="MathJax-Span-602"><span class="mi" id="MathJax-Span-603" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(2.92em, 1015em, 4.408em, -999.997em); top: -3.152em; left: 0em;"><span class="mtd" id="MathJax-Span-618"><span class="mrow" id="MathJax-Span-619"><span class="mi" id="MathJax-Span-620"></span><span class="texatom" id="MathJax-Span-621"><span class="mrow" id="MathJax-Span-622"></span></span><span class="mo" id="MathJax-Span-623" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="texatom" id="MathJax-Span-624" style="padding-left: 0.301em;"><span class="mrow" id="MathJax-Span-625"><span class="mi" id="MathJax-Span-626" style="font-family: MathJax_Main-bold;">V</span></span></span><span class="msubsup" id="MathJax-Span-627"><span style="display: inline-block; position: relative; width: 1.432em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.78em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-628"><span class="mrow" id="MathJax-Span-629"><span class="mi" id="MathJax-Span-630" style="font-family: MathJax_Main-bold;">Σ</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.836em;"><span class="texatom" id="MathJax-Span-631"><span class="mrow" id="MathJax-Span-632"><span class="texatom" id="MathJax-Span-633"><span class="mrow" id="MathJax-Span-634"><span class="mi" id="MathJax-Span-635" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-636"><span style="display: inline-block; position: relative; width: 1.491em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.84em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-637"><span class="mrow" id="MathJax-Span-638"><span class="mi" id="MathJax-Span-639" style="font-family: MathJax_Main-bold;">U</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.896em;"><span class="texatom" id="MathJax-Span-640"><span class="mrow" id="MathJax-Span-641"><span class="texatom" id="MathJax-Span-642"><span class="mrow" id="MathJax-Span-643"><span class="mi" id="MathJax-Span-644" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="texatom" id="MathJax-Span-645"><span class="mrow" id="MathJax-Span-646"><span class="mi" id="MathJax-Span-647" style="font-family: MathJax_Main-bold;">U</span></span></span><span class="texatom" id="MathJax-Span-648"><span class="mrow" id="MathJax-Span-649"><span class="mi" id="MathJax-Span-650" style="font-family: MathJax_Main-bold;">Σ</span></span></span><span class="msubsup" id="MathJax-Span-651"><span style="display: inline-block; position: relative; width: 1.432em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.84em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-652"><span class="mrow" id="MathJax-Span-653"><span class="mi" id="MathJax-Span-654" style="font-family: MathJax_Main-bold;">V</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.896em;"><span class="texatom" id="MathJax-Span-655"><span class="mrow" id="MathJax-Span-656"><span class="texatom" id="MathJax-Span-657"><span class="mrow" id="MathJax-Span-658"><span class="mi" id="MathJax-Span-659" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-660" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="texatom" id="MathJax-Span-661" style="padding-left: 0.301em;"><span class="mrow" id="MathJax-Span-662"><span class="mi" id="MathJax-Span-663" style="font-family: MathJax_Main-bold;">V</span></span></span><span class="mo" id="MathJax-Span-664" style="font-family: MathJax_Main;">(</span><span class="msubsup" id="MathJax-Span-665"><span style="display: inline-block; position: relative; width: 1.432em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.78em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-666"><span class="mrow" id="MathJax-Span-667"><span class="mi" id="MathJax-Span-668" style="font-family: MathJax_Main-bold;">Σ</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.836em;"><span class="texatom" id="MathJax-Span-669"><span class="mrow" id="MathJax-Span-670"><span class="texatom" id="MathJax-Span-671"><span class="mrow" id="MathJax-Span-672"><span class="mi" id="MathJax-Span-673" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="texatom" id="MathJax-Span-674"><span class="mrow" id="MathJax-Span-675"><span class="mi" id="MathJax-Span-676" style="font-family: MathJax_Main-bold;">Σ</span></span></span><span class="mo" id="MathJax-Span-677" style="font-family: MathJax_Main;">)</span><span class="msubsup" id="MathJax-Span-678"><span style="display: inline-block; position: relative; width: 1.432em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.84em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-679"><span class="mrow" id="MathJax-Span-680"><span class="mi" id="MathJax-Span-681" style="font-family: MathJax_Main-bold;">V</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.896em;"><span class="texatom" id="MathJax-Span-682"><span class="mrow" id="MathJax-Span-683"><span class="texatom" id="MathJax-Span-684"><span class="mrow" id="MathJax-Span-685"><span class="mi" id="MathJax-Span-686" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.354em; border-left: 0px solid; width: 0px; height: 3.289em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup></mtd><mtd><mi></mi><mrow class="MJX-TeXAtom-ORD"></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">Σ</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">V</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">V</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">Σ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">Σ</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">Σ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mo stretchy="false">)</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup></mtd></mtr><mtr><mtd><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow></mtd><mtd><mi></mi><mrow class="MJX-TeXAtom-ORD"></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">V</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">Σ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">Σ</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">V</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">V</mi></mrow><mo stretchy="false">(</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">Σ</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">Σ</mi></mrow><mo stretchy="false">)</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">V</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-37">\begin{aligned}
\mathbf M\mathbf M^{\mathsf H} &{}= \mathbf U\mathbf\Sigma\mathbf V^{\mathsf H}\mathbf V\mathbf\Sigma^{\mathsf H}\mathbf U^{\mathsf H} = \mathbf U(\mathbf\Sigma\mathbf\Sigma^{\mathsf H})\mathbf U^{\mathsf H}\\
\mathbf M^{\mathsf H}\mathbf M &{}= \mathbf V\mathbf\Sigma^{\mathsf H}\mathbf U^{\mathsf H}\mathbf U\mathbf\Sigma\mathbf V^{\mathsf H} = \mathbf V(\mathbf\Sigma^{\mathsf H}\mathbf\Sigma)\mathbf V^{\mathsf H}\\
\end{aligned}</script><p class=" has-jax has-jax has-jax has-jax has-jax has-jax has-jax has-jax">这也就是说，<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-38-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-687" style="width: 1.074em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.896em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1000.84em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-688"><span class="texatom" id="MathJax-Span-689"><span class="mrow" id="MathJax-Span-690"><span class="mi" id="MathJax-Span-691" style="font-family: MathJax_Main-bold;">U</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-38">\mathbf U</script> 的列向量（左奇异向量），是 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-39-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-692" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1002.74em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-693"><span class="texatom" id="MathJax-Span-694"><span class="mrow" id="MathJax-Span-695"><span class="mi" id="MathJax-Span-696" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="msubsup" id="MathJax-Span-697"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-698"><span class="mrow" id="MathJax-Span-699"><span class="mi" id="MathJax-Span-700" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-701"><span class="mrow" id="MathJax-Span-702"><span class="texatom" id="MathJax-Span-703"><span class="mrow" id="MathJax-Span-704"><span class="mi" id="MathJax-Span-705" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-39">\mathbf M\mathbf M^{\mathsf H}</script> 的特征向量；同时，<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-40-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-706" style="width: 1.074em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.896em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1000.9em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-707"><span class="texatom" id="MathJax-Span-708"><span class="mrow" id="MathJax-Span-709"><span class="mi" id="MathJax-Span-710" style="font-family: MathJax_Main-bold;">V</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">V</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-40">\mathbf V</script> 的列向量（右奇异向量），是 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-41-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-711" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1002.68em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-712"><span class="msubsup" id="MathJax-Span-713"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-714"><span class="mrow" id="MathJax-Span-715"><span class="mi" id="MathJax-Span-716" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-717"><span class="mrow" id="MathJax-Span-718"><span class="texatom" id="MathJax-Span-719"><span class="mrow" id="MathJax-Span-720"><span class="mi" id="MathJax-Span-721" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="texatom" id="MathJax-Span-722"><span class="mrow" id="MathJax-Span-723"><span class="mi" id="MathJax-Span-724" style="font-family: MathJax_Main-bold;">M</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-41">\mathbf M^{\mathsf H}\mathbf M</script> 的特征向量；另一方面，<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-42-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-725" style="width: 1.313em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.074em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1001.01em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-726"><span class="texatom" id="MathJax-Span-727"><span class="mrow" id="MathJax-Span-728"><span class="mi" id="MathJax-Span-729" style="font-family: MathJax_Main-bold;">M</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-42">\mathbf M</script> 的奇异值（<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-43-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;&amp;#x03A3;&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-730" style="width: 1.015em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.836em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1000.78em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-731"><span class="texatom" id="MathJax-Span-732"><span class="mrow" id="MathJax-Span-733"><span class="mi" id="MathJax-Span-734" style="font-family: MathJax_Main-bold;">Σ</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">Σ</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-43">\mathbf\Sigma</script> 的非零对角元素）则是 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-44-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-735" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1002.74em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-736"><span class="texatom" id="MathJax-Span-737"><span class="mrow" id="MathJax-Span-738"><span class="mi" id="MathJax-Span-739" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="msubsup" id="MathJax-Span-740"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-741"><span class="mrow" id="MathJax-Span-742"><span class="mi" id="MathJax-Span-743" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-744"><span class="mrow" id="MathJax-Span-745"><span class="texatom" id="MathJax-Span-746"><span class="mrow" id="MathJax-Span-747"><span class="mi" id="MathJax-Span-748" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-44">\mathbf M\mathbf M^{\mathsf H}</script> 或者 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-45-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-749" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1002.68em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-750"><span class="msubsup" id="MathJax-Span-751"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-752"><span class="mrow" id="MathJax-Span-753"><span class="mi" id="MathJax-Span-754" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-755"><span class="mrow" id="MathJax-Span-756"><span class="texatom" id="MathJax-Span-757"><span class="mrow" id="MathJax-Span-758"><span class="mi" id="MathJax-Span-759" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="texatom" id="MathJax-Span-760"><span class="mrow" id="MathJax-Span-761"><span class="mi" id="MathJax-Span-762" style="font-family: MathJax_Main-bold;">M</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-45">\mathbf M^{\mathsf H}\mathbf M</script> 的非零特征值的平方根。</p>
<h2 id="如何计算-SVD"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E5%A6%82%E4%BD%95%E8%AE%A1%E7%AE%97-SVD" class="headerlink" title="如何计算 SVD"></a>如何计算 SVD</h2><p>有了这些知识，我们就能手工计算出任意矩阵的 SVD 分解了；具体来说，算法如下</p>
<ol>
<li class=" has-jax has-jax">计算 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-46-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-763" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1002.74em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-764"><span class="texatom" id="MathJax-Span-765"><span class="mrow" id="MathJax-Span-766"><span class="mi" id="MathJax-Span-767" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="msubsup" id="MathJax-Span-768"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-769"><span class="mrow" id="MathJax-Span-770"><span class="mi" id="MathJax-Span-771" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-772"><span class="mrow" id="MathJax-Span-773"><span class="texatom" id="MathJax-Span-774"><span class="mrow" id="MathJax-Span-775"><span class="mi" id="MathJax-Span-776" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-46">\mathbf M\mathbf M^{\mathsf H}</script> 和 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-47-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-777" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1002.68em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-778"><span class="msubsup" id="MathJax-Span-779"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-780"><span class="mrow" id="MathJax-Span-781"><span class="mi" id="MathJax-Span-782" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-783"><span class="mrow" id="MathJax-Span-784"><span class="texatom" id="MathJax-Span-785"><span class="mrow" id="MathJax-Span-786"><span class="mi" id="MathJax-Span-787" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="texatom" id="MathJax-Span-788"><span class="mrow" id="MathJax-Span-789"><span class="mi" id="MathJax-Span-790" style="font-family: MathJax_Main-bold;">M</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-47">\mathbf M^{\mathsf H}\mathbf M</script>；</li>
<li class=" has-jax has-jax">分别计算 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-48-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-791" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1002.74em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-792"><span class="texatom" id="MathJax-Span-793"><span class="mrow" id="MathJax-Span-794"><span class="mi" id="MathJax-Span-795" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="msubsup" id="MathJax-Span-796"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-797"><span class="mrow" id="MathJax-Span-798"><span class="mi" id="MathJax-Span-799" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-800"><span class="mrow" id="MathJax-Span-801"><span class="texatom" id="MathJax-Span-802"><span class="mrow" id="MathJax-Span-803"><span class="mi" id="MathJax-Span-804" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-48">\mathbf M\mathbf M^{\mathsf H}</script> 和 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-49-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-805" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1002.68em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-806"><span class="msubsup" id="MathJax-Span-807"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-808"><span class="mrow" id="MathJax-Span-809"><span class="mi" id="MathJax-Span-810" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-811"><span class="mrow" id="MathJax-Span-812"><span class="texatom" id="MathJax-Span-813"><span class="mrow" id="MathJax-Span-814"><span class="mi" id="MathJax-Span-815" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="texatom" id="MathJax-Span-816"><span class="mrow" id="MathJax-Span-817"><span class="mi" id="MathJax-Span-818" style="font-family: MathJax_Main-bold;">M</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-49">\mathbf M^{\mathsf H}\mathbf M</script> 的特征向量及其特征值；</li>
<li class=" has-jax has-jax has-jax has-jax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-50-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-819" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1002.74em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-820"><span class="texatom" id="MathJax-Span-821"><span class="mrow" id="MathJax-Span-822"><span class="mi" id="MathJax-Span-823" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="msubsup" id="MathJax-Span-824"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-825"><span class="mrow" id="MathJax-Span-826"><span class="mi" id="MathJax-Span-827" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-828"><span class="mrow" id="MathJax-Span-829"><span class="texatom" id="MathJax-Span-830"><span class="mrow" id="MathJax-Span-831"><span class="mi" id="MathJax-Span-832" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-50">\mathbf M\mathbf M^{\mathsf H}</script> 的特征向量组成 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-51-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-833" style="width: 1.074em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.896em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1000.84em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-834"><span class="texatom" id="MathJax-Span-835"><span class="mrow" id="MathJax-Span-836"><span class="mi" id="MathJax-Span-837" style="font-family: MathJax_Main-bold;">U</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-51">\mathbf U</script>；而 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-52-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-838" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1002.68em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-839"><span class="msubsup" id="MathJax-Span-840"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-841"><span class="mrow" id="MathJax-Span-842"><span class="mi" id="MathJax-Span-843" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-844"><span class="mrow" id="MathJax-Span-845"><span class="texatom" id="MathJax-Span-846"><span class="mrow" id="MathJax-Span-847"><span class="mi" id="MathJax-Span-848" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="texatom" id="MathJax-Span-849"><span class="mrow" id="MathJax-Span-850"><span class="mi" id="MathJax-Span-851" style="font-family: MathJax_Main-bold;">M</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-52">\mathbf M^{\mathsf H}\mathbf M</script> 的特征向量组成 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-53-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-852" style="width: 1.074em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.896em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1000.9em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-853"><span class="texatom" id="MathJax-Span-854"><span class="mrow" id="MathJax-Span-855"><span class="mi" id="MathJax-Span-856" style="font-family: MathJax_Main-bold;">V</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">V</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-53">\mathbf V</script>；</li>
<li class=" has-jax has-jax has-jax">对 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-54-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-857" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1002.74em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-858"><span class="texatom" id="MathJax-Span-859"><span class="mrow" id="MathJax-Span-860"><span class="mi" id="MathJax-Span-861" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="msubsup" id="MathJax-Span-862"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-863"><span class="mrow" id="MathJax-Span-864"><span class="mi" id="MathJax-Span-865" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-866"><span class="mrow" id="MathJax-Span-867"><span class="texatom" id="MathJax-Span-868"><span class="mrow" id="MathJax-Span-869"><span class="mi" id="MathJax-Span-870" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-54">\mathbf M\mathbf M^{\mathsf H}</script> 和 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-55-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-871" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1002.68em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-872"><span class="msubsup" id="MathJax-Span-873"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-874"><span class="mrow" id="MathJax-Span-875"><span class="mi" id="MathJax-Span-876" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-877"><span class="mrow" id="MathJax-Span-878"><span class="texatom" id="MathJax-Span-879"><span class="mrow" id="MathJax-Span-880"><span class="mi" id="MathJax-Span-881" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="texatom" id="MathJax-Span-882"><span class="mrow" id="MathJax-Span-883"><span class="mi" id="MathJax-Span-884" style="font-family: MathJax_Main-bold;">M</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-55">\mathbf M^{\mathsf H}\mathbf M</script> 的非零特征值求平方根，对应上述特征向量的位置，填入 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-56-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;&amp;#x03A3;&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-885" style="width: 1.015em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.836em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1000.78em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-886"><span class="texatom" id="MathJax-Span-887"><span class="mrow" id="MathJax-Span-888"><span class="mi" id="MathJax-Span-889" style="font-family: MathJax_Main-bold;">Σ</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">Σ</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-56">\mathbf\Sigma</script> 的对角元。</li>
</ol>
<h2 id="实际计算看看"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E5%AE%9E%E9%99%85%E8%AE%A1%E7%AE%97%E7%9C%8B%E7%9C%8B" class="headerlink" title="实际计算看看"></a>实际计算看看</h2><p class=" has-jax has-jax has-jax has-jax">现在，我们来试着计算 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-57-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-890" style="width: 7.443em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.193em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(2.443em, 1005.9em, 7.979em, -999.997em); top: -5.473em; left: 0em;"><span class="mrow" id="MathJax-Span-891"><span class="texatom" id="MathJax-Span-892"><span class="mrow" id="MathJax-Span-893"><span class="mi" id="MathJax-Span-894" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="mo" id="MathJax-Span-895" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="mrow" id="MathJax-Span-896" style="padding-left: 0.301em;"><span class="mo" id="MathJax-Span-897" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎡<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.598em; left: 0em;">⎣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.664em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.068em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.533em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mtable" id="MathJax-Span-898" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 2.027em; height: 0px;"><span style="position: absolute; clip: rect(2.979em, 1000.48em, 8.217em, -999.997em); top: -5.89em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-899"><span class="mrow" id="MathJax-Span-900"><span class="mn" id="MathJax-Span-901" style="font-family: MathJax_Main;">2</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-905"><span class="mrow" id="MathJax-Span-906"><span class="mn" id="MathJax-Span-907" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-911"><span class="mrow" id="MathJax-Span-912"><span class="mn" id="MathJax-Span-913" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-917"><span class="mrow" id="MathJax-Span-918"><span class="mn" id="MathJax-Span-919" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1000.48em, 8.217em, -999.997em); top: -5.89em; left: 1.491em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-902"><span class="mrow" id="MathJax-Span-903"><span class="mn" id="MathJax-Span-904" style="font-family: MathJax_Main;">4</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-908"><span class="mrow" id="MathJax-Span-909"><span class="mn" id="MathJax-Span-910" style="font-family: MathJax_Main;">3</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-914"><span class="mrow" id="MathJax-Span-915"><span class="mn" id="MathJax-Span-916" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-920"><span class="mrow" id="MathJax-Span-921"><span class="mn" id="MathJax-Span-922" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span></span></span><span class="mo" id="MathJax-Span-923" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎤<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.598em; left: 0em;">⎦<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.664em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.068em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.533em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 5.479em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.854em; border-left: 0px solid; width: 0px; height: 6.361em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mo>=</mo><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>2</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo>]</mo></mrow></math></span></span><script type="math/tex" id="MathJax-Element-57">\mathbf M = \begin{bmatrix}2 & 4 \\ 1 & 3 \\ 0 & 0 \\ 0 & 0\end{bmatrix}</script> 的奇异值分解。计算奇异值分解，需要计算 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-58-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-924" style="width: 1.313em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.074em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1001.01em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-925"><span class="texatom" id="MathJax-Span-926"><span class="mrow" id="MathJax-Span-927"><span class="mi" id="MathJax-Span-928" style="font-family: MathJax_Main-bold;">M</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-58">\mathbf M</script> 与其共轭转置的左右积；这里主要以 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-59-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-929" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1002.74em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-930"><span class="texatom" id="MathJax-Span-931"><span class="mrow" id="MathJax-Span-932"><span class="mi" id="MathJax-Span-933" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="msubsup" id="MathJax-Span-934"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-935"><span class="mrow" id="MathJax-Span-936"><span class="mi" id="MathJax-Span-937" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-938"><span class="mrow" id="MathJax-Span-939"><span class="texatom" id="MathJax-Span-940"><span class="mrow" id="MathJax-Span-941"><span class="mi" id="MathJax-Span-942" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-59">\mathbf M\mathbf M^{\mathsf H}</script> 为例。<br>首先，我们需要计算 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-60-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-943" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1002.74em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-944"><span class="texatom" id="MathJax-Span-945"><span class="mrow" id="MathJax-Span-946"><span class="mi" id="MathJax-Span-947" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="msubsup" id="MathJax-Span-948"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-949"><span class="mrow" id="MathJax-Span-950"><span class="mi" id="MathJax-Span-951" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-952"><span class="mrow" id="MathJax-Span-953"><span class="texatom" id="MathJax-Span-954"><span class="mrow" id="MathJax-Span-955"><span class="mi" id="MathJax-Span-956" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-60">\mathbf M\mathbf M^{\mathsf H}</script>，</p>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-61-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;W&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;20&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;14&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;14&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-957" style="width: 31.908em; display: inline-block;"><span style="display: inline-block; position: relative; width: 26.551em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(2.979em, 1026.49em, 8.515em, -999.997em); top: -6.009em; left: 0em;"><span class="mrow" id="MathJax-Span-958"><span class="texatom" id="MathJax-Span-959"><span class="mrow" id="MathJax-Span-960"><span class="mi" id="MathJax-Span-961" style="font-family: MathJax_Main-bold;">W</span></span></span><span class="mo" id="MathJax-Span-962" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="texatom" id="MathJax-Span-963" style="padding-left: 0.301em;"><span class="mrow" id="MathJax-Span-964"><span class="mi" id="MathJax-Span-965" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="msubsup" id="MathJax-Span-966"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-967"><span class="mrow" id="MathJax-Span-968"><span class="mi" id="MathJax-Span-969" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-970"><span class="mrow" id="MathJax-Span-971"><span class="texatom" id="MathJax-Span-972"><span class="mrow" id="MathJax-Span-973"><span class="mi" id="MathJax-Span-974" style="font-size: 70.7%; font-family: MathJax_SansSerif;">H</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-975" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="mrow" id="MathJax-Span-976" style="padding-left: 0.301em;"><span class="mo" id="MathJax-Span-977" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎡<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.598em; left: 0em;">⎣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.664em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.068em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.533em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mtable" id="MathJax-Span-978" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 2.027em; height: 0px;"><span style="position: absolute; clip: rect(2.979em, 1000.48em, 8.217em, -999.997em); top: -5.89em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-979"><span class="mrow" id="MathJax-Span-980"><span class="mn" id="MathJax-Span-981" style="font-family: MathJax_Main;">2</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-985"><span class="mrow" id="MathJax-Span-986"><span class="mn" id="MathJax-Span-987" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-991"><span class="mrow" id="MathJax-Span-992"><span class="mn" id="MathJax-Span-993" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-997"><span class="mrow" id="MathJax-Span-998"><span class="mn" id="MathJax-Span-999" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1000.48em, 8.217em, -999.997em); top: -5.89em; left: 1.491em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-982"><span class="mrow" id="MathJax-Span-983"><span class="mn" id="MathJax-Span-984" style="font-family: MathJax_Main;">4</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-988"><span class="mrow" id="MathJax-Span-989"><span class="mn" id="MathJax-Span-990" style="font-family: MathJax_Main;">3</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-994"><span class="mrow" id="MathJax-Span-995"><span class="mn" id="MathJax-Span-996" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1000"><span class="mrow" id="MathJax-Span-1001"><span class="mn" id="MathJax-Span-1002" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span></span></span><span class="mo" id="MathJax-Span-1003" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎤<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.598em; left: 0em;">⎦<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.664em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.068em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.533em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span class="mrow" id="MathJax-Span-1004" style="padding-left: 0.182em;"><span class="mo" id="MathJax-Span-1005" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mtable" id="MathJax-Span-1006" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 5.003em; height: 0px;"><span style="position: absolute; clip: rect(2.503em, 1000.48em, 4.943em, -999.997em); top: -3.985em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1007"><span class="mrow" id="MathJax-Span-1008"><span class="mn" id="MathJax-Span-1009" style="font-family: MathJax_Main;">2</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1019"><span class="mrow" id="MathJax-Span-1020"><span class="mn" id="MathJax-Span-1021" style="font-family: MathJax_Main;">4</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(2.503em, 1000.48em, 4.943em, -999.997em); top: -3.985em; left: 1.491em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1010"><span class="mrow" id="MathJax-Span-1011"><span class="mn" id="MathJax-Span-1012" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1022"><span class="mrow" id="MathJax-Span-1023"><span class="mn" id="MathJax-Span-1024" style="font-family: MathJax_Main;">3</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(2.503em, 1000.48em, 4.943em, -999.997em); top: -3.985em; left: 2.979em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1013"><span class="mrow" id="MathJax-Span-1014"><span class="mn" id="MathJax-Span-1015" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1025"><span class="mrow" id="MathJax-Span-1026"><span class="mn" id="MathJax-Span-1027" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(2.503em, 1000.48em, 4.943em, -999.997em); top: -3.985em; left: 4.527em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1016"><span class="mrow" id="MathJax-Span-1017"><span class="mn" id="MathJax-Span-1018" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1028"><span class="mrow" id="MathJax-Span-1029"><span class="mn" id="MathJax-Span-1030" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-1031" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">]</span></span></span><span class="mo" id="MathJax-Span-1032" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="mrow" id="MathJax-Span-1033" style="padding-left: 0.301em;"><span class="mo" id="MathJax-Span-1034" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎡<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.598em; left: 0em;">⎣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.664em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.068em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.533em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mtable" id="MathJax-Span-1035" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 6.015em; height: 0px;"><span style="position: absolute; clip: rect(2.979em, 1000.96em, 8.217em, -999.997em); top: -5.89em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.96em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.473em;"><span class="mtd" id="MathJax-Span-1036"><span class="mrow" id="MathJax-Span-1037"><span class="mn" id="MathJax-Span-1038" style="font-family: MathJax_Main;">20</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.96em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.473em;"><span class="mtd" id="MathJax-Span-1048"><span class="mrow" id="MathJax-Span-1049"><span class="mn" id="MathJax-Span-1050" style="font-family: MathJax_Main;">14</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1060"><span class="mrow" id="MathJax-Span-1061"><span class="mn" id="MathJax-Span-1062" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1072"><span class="mrow" id="MathJax-Span-1073"><span class="mn" id="MathJax-Span-1074" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1000.96em, 8.217em, -999.997em); top: -5.89em; left: 2.027em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.96em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.473em;"><span class="mtd" id="MathJax-Span-1039"><span class="mrow" id="MathJax-Span-1040"><span class="mn" id="MathJax-Span-1041" style="font-family: MathJax_Main;">14</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.96em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.473em;"><span class="mtd" id="MathJax-Span-1051"><span class="mrow" id="MathJax-Span-1052"><span class="mn" id="MathJax-Span-1053" style="font-family: MathJax_Main;">10</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1063"><span class="mrow" id="MathJax-Span-1064"><span class="mn" id="MathJax-Span-1065" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1075"><span class="mrow" id="MathJax-Span-1076"><span class="mn" id="MathJax-Span-1077" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1000.48em, 8.217em, -999.997em); top: -5.89em; left: 3.991em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1042"><span class="mrow" id="MathJax-Span-1043"><span class="mn" id="MathJax-Span-1044" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1054"><span class="mrow" id="MathJax-Span-1055"><span class="mn" id="MathJax-Span-1056" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1066"><span class="mrow" id="MathJax-Span-1067"><span class="mn" id="MathJax-Span-1068" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1078"><span class="mrow" id="MathJax-Span-1079"><span class="mn" id="MathJax-Span-1080" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1000.48em, 8.217em, -999.997em); top: -5.89em; left: 5.479em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1045"><span class="mrow" id="MathJax-Span-1046"><span class="mn" id="MathJax-Span-1047" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1057"><span class="mrow" id="MathJax-Span-1058"><span class="mn" id="MathJax-Span-1059" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1069"><span class="mrow" id="MathJax-Span-1070"><span class="mn" id="MathJax-Span-1071" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1081"><span class="mrow" id="MathJax-Span-1082"><span class="mn" id="MathJax-Span-1083" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span></span></span><span class="mo" id="MathJax-Span-1084" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎤<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.598em; left: 0em;">⎦<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.664em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.068em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.533em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span class="mo" id="MathJax-Span-1085" style="font-family: MathJax_Main; padding-left: 0.182em;">.</span></span><span style="display: inline-block; width: 0px; height: 6.015em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.854em; border-left: 0px solid; width: 0px; height: 6.361em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">W</mi></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">H</mi></mrow></mrow></msup><mo>=</mo><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>2</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo>]</mo></mrow><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mn>3</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo>]</mo></mrow><mo>=</mo><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>20</mn></mtd><mtd><mn>14</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>14</mn></mtd><mtd><mn>10</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo>]</mo></mrow><mo>.</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-61">
\mathbf W = \mathbf M\mathbf M^{\mathsf H} = \begin{bmatrix}2 & 4 \\ 1 & 3 \\ 0 & 0 \\ 0 & 0\end{bmatrix}\begin{bmatrix}2 & 1 & 0 & 0 \\ 4 & 3 & 0 & 0\end{bmatrix} = \begin{bmatrix}20 & 14 & 0 & 0 \\ 14 & 10 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{bmatrix}.</script><p class=" has-jax has-jax has-jax">现在，我们要求 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-62-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;W&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1086" style="width: 1.432em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.193em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1001.19em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1087"><span class="texatom" id="MathJax-Span-1088"><span class="mrow" id="MathJax-Span-1089"><span class="mi" id="MathJax-Span-1090" style="font-family: MathJax_Main-bold;">W</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">W</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-62">\mathbf W</script> 的特征值与特征向量。根据定义 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-63-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;W&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x2192;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x2192;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1091" style="width: 5.241em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.348em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.67em, 1004.29em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1092"><span class="texatom" id="MathJax-Span-1093"><span class="mrow" id="MathJax-Span-1094"><span class="mi" id="MathJax-Span-1095" style="font-family: MathJax_Main-bold;">W</span></span></span><span class="texatom" id="MathJax-Span-1096"><span class="mrow" id="MathJax-Span-1097"><span class="munderover" id="MathJax-Span-1098"><span style="display: inline-block; position: relative; width: 0.598em; height: 0px;"><span style="position: absolute; clip: rect(3.396em, 1000.54em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-1099" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.045em; left: 0.122em;"><span style="height: 0em; vertical-align: 0em; width: 0.479em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-1100" style="font-family: MathJax_Main;">⃗&nbsp;<span style="height: 0em; vertical-align: 0em; margin-left: -0.235em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1101" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="mi" id="MathJax-Span-1102" style="font-family: MathJax_Math-italic; padding-left: 0.301em;">λ</span><span class="texatom" id="MathJax-Span-1103"><span class="mrow" id="MathJax-Span-1104"><span class="munderover" id="MathJax-Span-1105"><span style="display: inline-block; position: relative; width: 0.598em; height: 0px;"><span style="position: absolute; clip: rect(3.396em, 1000.54em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-1106" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.045em; left: 0.122em;"><span style="height: 0em; vertical-align: 0em; width: 0.479em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-1107" style="font-family: MathJax_Main;">⃗&nbsp;<span style="height: 0em; vertical-align: 0em; margin-left: -0.235em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.075em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">W</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mi>λ</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-63">\mathbf W\vec x = \lambda \vec x</script>；因此 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-64-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;W&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x2192;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x2192;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1108" style="width: 8.098em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.729em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.432em, 1006.67em, 3.039em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1109"><span class="mo" id="MathJax-Span-1110" style="font-family: MathJax_Main;">(</span><span class="texatom" id="MathJax-Span-1111"><span class="mrow" id="MathJax-Span-1112"><span class="mi" id="MathJax-Span-1113" style="font-family: MathJax_Main-bold;">W</span></span></span><span class="mo" id="MathJax-Span-1114" style="font-family: MathJax_Main; padding-left: 0.241em;">−</span><span class="mi" id="MathJax-Span-1115" style="font-family: MathJax_Math-italic; padding-left: 0.241em;">λ</span><span class="texatom" id="MathJax-Span-1116"><span class="mrow" id="MathJax-Span-1117"><span class="mi" id="MathJax-Span-1118" style="font-family: MathJax_Main-bold;">I</span></span></span><span class="mo" id="MathJax-Span-1119" style="font-family: MathJax_Main;">)</span><span class="texatom" id="MathJax-Span-1120"><span class="mrow" id="MathJax-Span-1121"><span class="munderover" id="MathJax-Span-1122"><span style="display: inline-block; position: relative; width: 0.598em; height: 0px;"><span style="position: absolute; clip: rect(3.396em, 1000.54em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-1123" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.045em; left: 0.122em;"><span style="height: 0em; vertical-align: 0em; width: 0.479em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-1124" style="font-family: MathJax_Main;">⃗&nbsp;<span style="height: 0em; vertical-align: 0em; margin-left: -0.235em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1125" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="texatom" id="MathJax-Span-1126" style="padding-left: 0.301em;"><span class="mrow" id="MathJax-Span-1127"><span class="munderover" id="MathJax-Span-1128"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mn" id="MathJax-Span-1129" style="font-family: MathJax_Main;">0</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.283em; left: 0.003em;"><span style="height: 0em; vertical-align: 0em; width: 0.479em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-1130" style="font-family: MathJax_Main;">⃗&nbsp;<span style="height: 0em; vertical-align: 0em; margin-left: -0.235em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left: 0px solid; width: 0px; height: 1.646em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">W</mi></mrow><mo>−</mo><mi>λ</mi><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">I</mi></mrow><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mover><mn>0</mn><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-64">(\mathbf W - \lambda\mathbf I)\vec x = \vec 0</script>。这也就是说</p>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-65-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;20&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;14&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;14&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x2192;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x2192;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1131" style="width: 19.229em; display: inline-block;"><span style="display: inline-block; position: relative; width: 16.015em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(2.443em, 1015.96em, 7.979em, -999.997em); top: -5.473em; left: 0em;"><span class="mrow" id="MathJax-Span-1132"><span class="mrow" id="MathJax-Span-1133"><span class="mo" id="MathJax-Span-1134" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎡<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.598em; left: 0em;">⎣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.664em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.068em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.533em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mtable" id="MathJax-Span-1135" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 11.432em; height: 0px;"><span style="position: absolute; clip: rect(2.979em, 1002.8em, 8.217em, -999.997em); top: -5.89em; left: 0em;"><span style="display: inline-block; position: relative; width: 2.86em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1002.8em, 4.229em, -999.997em); top: -6.009em; left: 50%; margin-left: -1.426em;"><span class="mtd" id="MathJax-Span-1136"><span class="mrow" id="MathJax-Span-1137"><span class="mn" id="MathJax-Span-1138" style="font-family: MathJax_Main;">20</span><span class="mo" id="MathJax-Span-1139" style="font-family: MathJax_Main; padding-left: 0.241em;">−</span><span class="mi" id="MathJax-Span-1140" style="font-family: MathJax_Math-italic; padding-left: 0.241em;">λ</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.96em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.473em;"><span class="mtd" id="MathJax-Span-1150"><span class="mrow" id="MathJax-Span-1151"><span class="mn" id="MathJax-Span-1152" style="font-family: MathJax_Main;">14</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1164"><span class="mrow" id="MathJax-Span-1165"><span class="mn" id="MathJax-Span-1166" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1177"><span class="mrow" id="MathJax-Span-1178"><span class="mn" id="MathJax-Span-1179" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1002.8em, 8.217em, -999.997em); top: -5.89em; left: 3.872em;"><span style="display: inline-block; position: relative; width: 2.86em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.96em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.473em;"><span class="mtd" id="MathJax-Span-1141"><span class="mrow" id="MathJax-Span-1142"><span class="mn" id="MathJax-Span-1143" style="font-family: MathJax_Main;">14</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.098em, 1002.8em, 4.229em, -999.997em); top: -4.64em; left: 50%; margin-left: -1.426em;"><span class="mtd" id="MathJax-Span-1153"><span class="mrow" id="MathJax-Span-1154"><span class="mn" id="MathJax-Span-1155" style="font-family: MathJax_Main;">10</span><span class="mo" id="MathJax-Span-1156" style="font-family: MathJax_Main; padding-left: 0.241em;">−</span><span class="mi" id="MathJax-Span-1157" style="font-family: MathJax_Math-italic; padding-left: 0.241em;">λ</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1167"><span class="mrow" id="MathJax-Span-1168"><span class="mn" id="MathJax-Span-1169" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1180"><span class="mrow" id="MathJax-Span-1181"><span class="mn" id="MathJax-Span-1182" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1001.31em, 8.217em, -999.997em); top: -5.89em; left: 7.682em;"><span style="display: inline-block; position: relative; width: 1.372em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1144"><span class="mrow" id="MathJax-Span-1145"><span class="mn" id="MathJax-Span-1146" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1158"><span class="mrow" id="MathJax-Span-1159"><span class="mn" id="MathJax-Span-1160" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.098em, 1001.31em, 4.229em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.652em;"><span class="mtd" id="MathJax-Span-1170"><span class="mrow" id="MathJax-Span-1171"><span class="mo" id="MathJax-Span-1172" style="font-family: MathJax_Main;">−</span><span class="mi" id="MathJax-Span-1173" style="font-family: MathJax_Math-italic;">λ</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1183"><span class="mrow" id="MathJax-Span-1184"><span class="mn" id="MathJax-Span-1185" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(3.039em, 1001.31em, 8.336em, -999.997em); top: -5.949em; left: 10.063em;"><span style="display: inline-block; position: relative; width: 1.372em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1147"><span class="mrow" id="MathJax-Span-1148"><span class="mn" id="MathJax-Span-1149" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1161"><span class="mrow" id="MathJax-Span-1162"><span class="mn" id="MathJax-Span-1163" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1174"><span class="mrow" id="MathJax-Span-1175"><span class="mn" id="MathJax-Span-1176" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.098em, 1001.31em, 4.229em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.652em;"><span class="mtd" id="MathJax-Span-1186"><span class="mrow" id="MathJax-Span-1187"><span class="mo" id="MathJax-Span-1188" style="font-family: MathJax_Main;">−</span><span class="mi" id="MathJax-Span-1189" style="font-family: MathJax_Math-italic;">λ</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.955em;"></span></span></span></span><span class="mo" id="MathJax-Span-1190" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎤<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.598em; left: 0em;">⎦<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.664em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.068em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.533em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span class="texatom" id="MathJax-Span-1191" style="padding-left: 0.182em;"><span class="mrow" id="MathJax-Span-1192"><span class="munderover" id="MathJax-Span-1193"><span style="display: inline-block; position: relative; width: 0.598em; height: 0px;"><span style="position: absolute; clip: rect(3.396em, 1000.54em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-1194" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.045em; left: 0.122em;"><span style="height: 0em; vertical-align: 0em; width: 0.479em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-1195" style="font-family: MathJax_Main;">⃗&nbsp;<span style="height: 0em; vertical-align: 0em; margin-left: -0.235em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1196" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="texatom" id="MathJax-Span-1197" style="padding-left: 0.301em;"><span class="mrow" id="MathJax-Span-1198"><span class="munderover" id="MathJax-Span-1199"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mn" id="MathJax-Span-1200" style="font-family: MathJax_Main;">0</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.283em; left: 0.003em;"><span style="height: 0em; vertical-align: 0em; width: 0.479em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-1201" style="font-family: MathJax_Main;">⃗&nbsp;<span style="height: 0em; vertical-align: 0em; margin-left: -0.235em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1202" style="font-family: MathJax_Main;">.</span></span><span style="display: inline-block; width: 0px; height: 5.479em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.854em; border-left: 0px solid; width: 0px; height: 6.361em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>20</mn><mo>−</mo><mi>λ</mi></mtd><mtd><mn>14</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>14</mn></mtd><mtd><mn>10</mn><mo>−</mo><mi>λ</mi></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mi>λ</mi></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mi>λ</mi></mtd></mtr></mtable><mo>]</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mover><mn>0</mn><mo stretchy="false">→</mo></mover></mrow><mo>.</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-65">
\begin{bmatrix}
20 - \lambda & 14 & 0 & 0 \\
14 & 10 - \lambda & 0 & 0 \\
0 & 0 & -\lambda & 0 \\
0 & 0 & 0 & -\lambda
\end{bmatrix}\vec x = \vec 0.</script><p class=" has-jax">根据线性方程组的理论，若要该关于 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-66-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x2192;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1203" style="width: 0.717em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.598em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.67em, 1000.54em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1204"><span class="texatom" id="MathJax-Span-1205"><span class="mrow" id="MathJax-Span-1206"><span class="munderover" id="MathJax-Span-1207"><span style="display: inline-block; position: relative; width: 0.598em; height: 0px;"><span style="position: absolute; clip: rect(3.396em, 1000.54em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-1208" style="font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.045em; left: 0.122em;"><span style="height: 0em; vertical-align: 0em; width: 0.479em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-1209" style="font-family: MathJax_Main;">⃗&nbsp;<span style="height: 0em; vertical-align: 0em; margin-left: -0.235em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.075em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-66">\vec x</script> 的方程有非零解，则要求系数矩阵的行列式为 0；也就是</p>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-67-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;20&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;14&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;14&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;20&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;14&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;14&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1210" style="width: 34.17em; display: inline-block;"><span style="display: inline-block; position: relative; width: 28.455em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(2.443em, 1028.4em, 7.979em, -999.997em); top: -5.473em; left: 0em;"><span class="mrow" id="MathJax-Span-1211"><span class="mrow" id="MathJax-Span-1212"><span class="mo" id="MathJax-Span-1213" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.301em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.211em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: 0.955em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -2.378em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -1.545em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -0.711em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: 0.122em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mtable" id="MathJax-Span-1214" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 11.432em; height: 0px;"><span style="position: absolute; clip: rect(2.979em, 1002.8em, 8.217em, -999.997em); top: -5.89em; left: 0em;"><span style="display: inline-block; position: relative; width: 2.86em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1002.8em, 4.229em, -999.997em); top: -6.009em; left: 50%; margin-left: -1.426em;"><span class="mtd" id="MathJax-Span-1215"><span class="mrow" id="MathJax-Span-1216"><span class="mn" id="MathJax-Span-1217" style="font-family: MathJax_Main;">20</span><span class="mo" id="MathJax-Span-1218" style="font-family: MathJax_Main; padding-left: 0.241em;">−</span><span class="mi" id="MathJax-Span-1219" style="font-family: MathJax_Math-italic; padding-left: 0.241em;">λ</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.96em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.473em;"><span class="mtd" id="MathJax-Span-1229"><span class="mrow" id="MathJax-Span-1230"><span class="mn" id="MathJax-Span-1231" style="font-family: MathJax_Main;">14</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1243"><span class="mrow" id="MathJax-Span-1244"><span class="mn" id="MathJax-Span-1245" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1256"><span class="mrow" id="MathJax-Span-1257"><span class="mn" id="MathJax-Span-1258" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1002.8em, 8.217em, -999.997em); top: -5.89em; left: 3.872em;"><span style="display: inline-block; position: relative; width: 2.86em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.96em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.473em;"><span class="mtd" id="MathJax-Span-1220"><span class="mrow" id="MathJax-Span-1221"><span class="mn" id="MathJax-Span-1222" style="font-family: MathJax_Main;">14</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.098em, 1002.8em, 4.229em, -999.997em); top: -4.64em; left: 50%; margin-left: -1.426em;"><span class="mtd" id="MathJax-Span-1232"><span class="mrow" id="MathJax-Span-1233"><span class="mn" id="MathJax-Span-1234" style="font-family: MathJax_Main;">10</span><span class="mo" id="MathJax-Span-1235" style="font-family: MathJax_Main; padding-left: 0.241em;">−</span><span class="mi" id="MathJax-Span-1236" style="font-family: MathJax_Math-italic; padding-left: 0.241em;">λ</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1246"><span class="mrow" id="MathJax-Span-1247"><span class="mn" id="MathJax-Span-1248" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1259"><span class="mrow" id="MathJax-Span-1260"><span class="mn" id="MathJax-Span-1261" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1001.31em, 8.217em, -999.997em); top: -5.89em; left: 7.682em;"><span style="display: inline-block; position: relative; width: 1.372em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1223"><span class="mrow" id="MathJax-Span-1224"><span class="mn" id="MathJax-Span-1225" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1237"><span class="mrow" id="MathJax-Span-1238"><span class="mn" id="MathJax-Span-1239" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.098em, 1001.31em, 4.229em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.652em;"><span class="mtd" id="MathJax-Span-1249"><span class="mrow" id="MathJax-Span-1250"><span class="mo" id="MathJax-Span-1251" style="font-family: MathJax_Main;">−</span><span class="mi" id="MathJax-Span-1252" style="font-family: MathJax_Math-italic;">λ</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1262"><span class="mrow" id="MathJax-Span-1263"><span class="mn" id="MathJax-Span-1264" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(3.039em, 1001.31em, 8.336em, -999.997em); top: -5.949em; left: 10.063em;"><span style="display: inline-block; position: relative; width: 1.372em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1226"><span class="mrow" id="MathJax-Span-1227"><span class="mn" id="MathJax-Span-1228" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1240"><span class="mrow" id="MathJax-Span-1241"><span class="mn" id="MathJax-Span-1242" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1253"><span class="mrow" id="MathJax-Span-1254"><span class="mn" id="MathJax-Span-1255" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.098em, 1001.31em, 4.229em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.652em;"><span class="mtd" id="MathJax-Span-1265"><span class="mrow" id="MathJax-Span-1266"><span class="mo" id="MathJax-Span-1267" style="font-family: MathJax_Main;">−</span><span class="mi" id="MathJax-Span-1268" style="font-family: MathJax_Math-italic;">λ</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.955em;"></span></span></span></span><span class="mo" id="MathJax-Span-1269" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.301em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.211em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: 0.955em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -2.378em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -1.545em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -0.711em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: 0.122em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span class="mo" id="MathJax-Span-1270" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="mrow" id="MathJax-Span-1271" style="padding-left: 0.301em;"><span class="mo" id="MathJax-Span-1272" style="vertical-align: 1.432em;"><span style="display: inline-block; position: relative; width: 0.301em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.211em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -1.842em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -2.557em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mtable" id="MathJax-Span-1273" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 6.67em; height: 0px;"><span style="position: absolute; clip: rect(2.443em, 1002.8em, 4.943em, -999.997em); top: -3.985em; left: 0em;"><span style="display: inline-block; position: relative; width: 2.86em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1002.8em, 4.229em, -999.997em); top: -4.64em; left: 50%; margin-left: -1.426em;"><span class="mtd" id="MathJax-Span-1274"><span class="mrow" id="MathJax-Span-1275"><span class="mn" id="MathJax-Span-1276" style="font-family: MathJax_Main;">20</span><span class="mo" id="MathJax-Span-1277" style="font-family: MathJax_Main; padding-left: 0.241em;">−</span><span class="mi" id="MathJax-Span-1278" style="font-family: MathJax_Math-italic; padding-left: 0.241em;">λ</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.96em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.473em;"><span class="mtd" id="MathJax-Span-1282"><span class="mrow" id="MathJax-Span-1283"><span class="mn" id="MathJax-Span-1284" style="font-family: MathJax_Main;">14</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(2.503em, 1002.8em, 5.003em, -999.997em); top: -3.985em; left: 3.872em;"><span style="display: inline-block; position: relative; width: 2.86em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.96em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.473em;"><span class="mtd" id="MathJax-Span-1279"><span class="mrow" id="MathJax-Span-1280"><span class="mn" id="MathJax-Span-1281" style="font-family: MathJax_Main;">14</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.098em, 1002.8em, 4.229em, -999.997em); top: -3.211em; left: 50%; margin-left: -1.426em;"><span class="mtd" id="MathJax-Span-1285"><span class="mrow" id="MathJax-Span-1286"><span class="mn" id="MathJax-Span-1287" style="font-family: MathJax_Main;">10</span><span class="mo" id="MathJax-Span-1288" style="font-family: MathJax_Main; padding-left: 0.241em;">−</span><span class="mi" id="MathJax-Span-1289" style="font-family: MathJax_Math-italic; padding-left: 0.241em;">λ</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-1290" style="vertical-align: 1.432em;"><span style="display: inline-block; position: relative; width: 0.301em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.211em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -1.842em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -2.557em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span class="mrow" id="MathJax-Span-1291" style="padding-left: 0.182em;"><span class="mo" id="MathJax-Span-1292" style="vertical-align: 1.432em;"><span style="display: inline-block; position: relative; width: 0.301em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.211em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -1.842em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -2.557em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mtable" id="MathJax-Span-1293" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 3.753em; height: 0px;"><span style="position: absolute; clip: rect(2.443em, 1001.31em, 4.943em, -999.997em); top: -3.985em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.372em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.31em, 4.229em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.652em;"><span class="mtd" id="MathJax-Span-1294"><span class="mrow" id="MathJax-Span-1295"><span class="mo" id="MathJax-Span-1296" style="font-family: MathJax_Main;">−</span><span class="mi" id="MathJax-Span-1297" style="font-family: MathJax_Math-italic;">λ</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1301"><span class="mrow" id="MathJax-Span-1302"><span class="mn" id="MathJax-Span-1303" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(2.503em, 1001.31em, 5.003em, -999.997em); top: -3.985em; left: 2.384em;"><span style="display: inline-block; position: relative; width: 1.372em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1298"><span class="mrow" id="MathJax-Span-1299"><span class="mn" id="MathJax-Span-1300" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.098em, 1001.31em, 4.229em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.652em;"><span class="mtd" id="MathJax-Span-1304"><span class="mrow" id="MathJax-Span-1305"><span class="mo" id="MathJax-Span-1306" style="font-family: MathJax_Main;">−</span><span class="mi" id="MathJax-Span-1307" style="font-family: MathJax_Math-italic;">λ</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-1308" style="vertical-align: 1.432em;"><span style="display: inline-block; position: relative; width: 0.301em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.211em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -1.842em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -2.557em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span class="mo" id="MathJax-Span-1309" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="mn" id="MathJax-Span-1310" style="font-family: MathJax_Main; padding-left: 0.301em;">0</span><span class="mo" id="MathJax-Span-1311" style="font-family: MathJax_Main;">,</span></span><span style="display: inline-block; width: 0px; height: 5.479em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.854em; border-left: 0px solid; width: 0px; height: 6.361em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>|</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>20</mn><mo>−</mo><mi>λ</mi></mtd><mtd><mn>14</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>14</mn></mtd><mtd><mn>10</mn><mo>−</mo><mi>λ</mi></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mi>λ</mi></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mi>λ</mi></mtd></mtr></mtable><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>20</mn><mo>−</mo><mi>λ</mi></mtd><mtd><mn>14</mn></mtd></mtr><mtr><mtd><mn>14</mn></mtd><mtd><mn>10</mn><mo>−</mo><mi>λ</mi></mtd></mtr></mtable><mo>|</mo></mrow><mrow><mo>|</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mo>−</mo><mi>λ</mi></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mi>λ</mi></mtd></mtr></mtable><mo>|</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-67">
\begin{vmatrix}
20 - \lambda & 14 & 0 & 0 \\
14 & 10 - \lambda & 0 & 0 \\
0 & 0 & -\lambda & 0 \\
0 & 0 & 0 & -\lambda
\end{vmatrix} =
\begin{vmatrix}
20 - \lambda & 14 \\
14 & 10 - \lambda \\
\end{vmatrix}\begin{vmatrix}
-\lambda & 0 \\
0 & -\lambda \\
\end{vmatrix}
= 0,</script><p class=" has-jax has-jax has-jax has-jax">这也就是 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-68-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;20&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;196&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1312" style="width: 16.61em; display: inline-block;"><span style="display: inline-block; position: relative; width: 13.813em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.61em, 1013.75em, 3.158em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1313"><span class="texatom" id="MathJax-Span-1314"><span class="mrow" id="MathJax-Span-1315"><span class="mo" id="MathJax-Span-1316" style="vertical-align: 0em;"><span style="font-family: MathJax_Size1;">(</span></span></span></span><span class="mo" id="MathJax-Span-1317" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-1318" style="font-family: MathJax_Main;">20</span><span class="mo" id="MathJax-Span-1319" style="font-family: MathJax_Main; padding-left: 0.241em;">−</span><span class="mi" id="MathJax-Span-1320" style="font-family: MathJax_Math-italic; padding-left: 0.241em;">λ</span><span class="mo" id="MathJax-Span-1321" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-1322" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-1323" style="font-family: MathJax_Main;">10</span><span class="mo" id="MathJax-Span-1324" style="font-family: MathJax_Main; padding-left: 0.241em;">−</span><span class="mi" id="MathJax-Span-1325" style="font-family: MathJax_Math-italic; padding-left: 0.241em;">λ</span><span class="mo" id="MathJax-Span-1326" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-1327" style="font-family: MathJax_Main; padding-left: 0.241em;">−</span><span class="mn" id="MathJax-Span-1328" style="font-family: MathJax_Main; padding-left: 0.241em;">196</span><span class="texatom" id="MathJax-Span-1329"><span class="mrow" id="MathJax-Span-1330"><span class="mo" id="MathJax-Span-1331" style="vertical-align: 0em;"><span style="font-family: MathJax_Size1;">)</span></span></span></span><span class="msubsup" id="MathJax-Span-1332"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.54em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-1333" style="font-family: MathJax_Math-italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.342em; left: 0.598em;"><span class="mn" id="MathJax-Span-1334" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-1335" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="mn" id="MathJax-Span-1336" style="font-family: MathJax_Main; padding-left: 0.301em;">0</span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.496em; border-left: 0px solid; width: 0px; height: 1.575em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mo stretchy="false">(</mo><mn>20</mn><mo>−</mo><mi>λ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>10</mn><mo>−</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>−</mo><mn>196</mn><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><msup><mi>λ</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-68">\bigl((20 - \lambda)(10 - \lambda) - 196\bigr)\lambda^2 = 0</script>；解得 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-69-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1337" style="width: 6.372em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.301em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1005.24em, 2.979em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1338"><span class="msubsup" id="MathJax-Span-1339"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.54em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-1340" style="font-family: MathJax_Math-italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.807em; left: 0.598em;"><span class="texatom" id="MathJax-Span-1341"><span class="mrow" id="MathJax-Span-1342"><span class="mn" id="MathJax-Span-1343" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-1344" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="msubsup" id="MathJax-Span-1345" style="padding-left: 0.301em;"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.54em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-1346" style="font-family: MathJax_Math-italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.807em; left: 0.598em;"><span class="texatom" id="MathJax-Span-1347"><span class="mrow" id="MathJax-Span-1348"><span class="mn" id="MathJax-Span-1349" style="font-size: 70.7%; font-family: MathJax_Main;">2</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-1350" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="mn" id="MathJax-Span-1351" style="font-family: MathJax_Main; padding-left: 0.301em;">0</span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.146em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-69">\lambda_{1} = \lambda_{2} = 0</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-70-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;15&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msqrt&gt;&lt;mn&gt;221&lt;/mn&gt;&lt;/msqrt&gt;&lt;mo&gt;&amp;#x2248;&lt;/mo&gt;&lt;mn&gt;29.866&lt;/mn&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1352" style="width: 13.455em; display: inline-block;"><span style="display: inline-block; position: relative; width: 11.193em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.61em, 1011.13em, 2.979em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1353"><span class="msubsup" id="MathJax-Span-1354"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.54em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-1355" style="font-family: MathJax_Math-italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.807em; left: 0.598em;"><span class="texatom" id="MathJax-Span-1356"><span class="mrow" id="MathJax-Span-1357"><span class="mn" id="MathJax-Span-1358" style="font-size: 70.7%; font-family: MathJax_Main;">3</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-1359" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="mn" id="MathJax-Span-1360" style="font-family: MathJax_Main; padding-left: 0.301em;">15</span><span class="mo" id="MathJax-Span-1361" style="font-family: MathJax_Main; padding-left: 0.241em;">+</span><span class="msqrt" id="MathJax-Span-1362" style="padding-left: 0.241em;"><span style="display: inline-block; position: relative; width: 2.384em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1001.43em, 4.17em, -999.997em); top: -3.985em; left: 0.836em;"><span class="mrow" id="MathJax-Span-1363"><span class="mn" id="MathJax-Span-1364" style="font-family: MathJax_Main;">221</span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.515em, 1001.55em, 3.932em, -999.997em); top: -4.521em; left: 0.836em;"><span style="display: inline-block; position: relative; width: 1.551em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.985em; left: -0.057em;">−<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -3.985em; left: 0.836em;">−<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -3.985em; left: 0.36em;">−<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.039em, 1000.84em, 4.348em, -999.997em); top: -4.045em; left: 0em;"><span style="font-family: MathJax_Main;">√</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-1365" style="font-family: MathJax_Main; padding-left: 0.301em;">≈</span><span class="mn" id="MathJax-Span-1366" style="font-family: MathJax_Main; padding-left: 0.301em;">29.866</span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.361em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mrow class="MJX-TeXAtom-ORD"><mn>3</mn></mrow></msub><mo>=</mo><mn>15</mn><mo>+</mo><msqrt><mn>221</mn></msqrt><mo>≈</mo><mn>29.866</mn></math></span></span><script type="math/tex" id="MathJax-Element-70">\lambda_{3} = 15 + \sqrt{221} \approx 29.866</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-71-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;15&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msqrt&gt;&lt;mn&gt;221&lt;/mn&gt;&lt;/msqrt&gt;&lt;mo&gt;&amp;#x2248;&lt;/mo&gt;&lt;mn&gt;0.134&lt;/mn&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1367" style="width: 12.86em; display: inline-block;"><span style="display: inline-block; position: relative; width: 10.717em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.61em, 1010.72em, 2.979em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1368"><span class="msubsup" id="MathJax-Span-1369"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1000.54em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-1370" style="font-family: MathJax_Math-italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.807em; left: 0.598em;"><span class="texatom" id="MathJax-Span-1371"><span class="mrow" id="MathJax-Span-1372"><span class="mn" id="MathJax-Span-1373" style="font-size: 70.7%; font-family: MathJax_Main;">4</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-1374" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="mn" id="MathJax-Span-1375" style="font-family: MathJax_Main; padding-left: 0.301em;">15</span><span class="mo" id="MathJax-Span-1376" style="font-family: MathJax_Main; padding-left: 0.241em;">−</span><span class="msqrt" id="MathJax-Span-1377" style="padding-left: 0.241em;"><span style="display: inline-block; position: relative; width: 2.384em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1001.43em, 4.17em, -999.997em); top: -3.985em; left: 0.836em;"><span class="mrow" id="MathJax-Span-1378"><span class="mn" id="MathJax-Span-1379" style="font-family: MathJax_Main;">221</span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.515em, 1001.55em, 3.932em, -999.997em); top: -4.521em; left: 0.836em;"><span style="display: inline-block; position: relative; width: 1.551em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.985em; left: -0.057em;">−<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -3.985em; left: 0.836em;">−<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -3.985em; left: 0.36em;">−<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.039em, 1000.84em, 4.348em, -999.997em); top: -4.045em; left: 0em;"><span style="font-family: MathJax_Main;">√</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-1380" style="font-family: MathJax_Main; padding-left: 0.301em;">≈</span><span class="mn" id="MathJax-Span-1381" style="font-family: MathJax_Main; padding-left: 0.301em;">0.134</span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.361em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mrow class="MJX-TeXAtom-ORD"><mn>4</mn></mrow></msub><mo>=</mo><mn>15</mn><mo>−</mo><msqrt><mn>221</mn></msqrt><mo>≈</mo><mn>0.134</mn></math></span></span><script type="math/tex" id="MathJax-Element-71">\lambda_{4} = 15 - \sqrt{221} \approx 0.134</script>。将特征值代入原方程，可解得对应的特征向量；这些特征向量即作为列向量，形成矩阵</p>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-72-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;0.82&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;0.58&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;0.58&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0.82&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1382" style="width: 16.253em; display: inline-block;"><span style="display: inline-block; position: relative; width: 13.515em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(2.443em, 1013.46em, 7.979em, -999.997em); top: -5.473em; left: 0em;"><span class="mrow" id="MathJax-Span-1383"><span class="texatom" id="MathJax-Span-1384"><span class="mrow" id="MathJax-Span-1385"><span class="mi" id="MathJax-Span-1386" style="font-family: MathJax_Main-bold;">U</span></span></span><span class="mo" id="MathJax-Span-1387" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="mrow" id="MathJax-Span-1388" style="padding-left: 0.301em;"><span class="mo" id="MathJax-Span-1389" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎡<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.598em; left: 0em;">⎣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.664em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.068em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.533em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mtable" id="MathJax-Span-1390" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 9.11em; height: 0px;"><span style="position: absolute; clip: rect(2.979em, 1002.5em, 8.217em, -999.997em); top: -5.89em; left: 0em;"><span style="display: inline-block; position: relative; width: 2.562em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1002.5em, 4.229em, -999.997em); top: -6.009em; left: 50%; margin-left: -1.247em;"><span class="mtd" id="MathJax-Span-1391"><span class="mrow" id="MathJax-Span-1392"><span class="mo" id="MathJax-Span-1393" style="font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-1394" style="font-family: MathJax_Main;">0.82</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1002.5em, 4.229em, -999.997em); top: -4.64em; left: 50%; margin-left: -1.247em;"><span class="mtd" id="MathJax-Span-1405"><span class="mrow" id="MathJax-Span-1406"><span class="mo" id="MathJax-Span-1407" style="font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-1408" style="font-family: MathJax_Main;">0.58</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1418"><span class="mrow" id="MathJax-Span-1419"><span class="mn" id="MathJax-Span-1420" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1430"><span class="mrow" id="MathJax-Span-1431"><span class="mn" id="MathJax-Span-1432" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1002.5em, 8.217em, -999.997em); top: -5.89em; left: 3.574em;"><span style="display: inline-block; position: relative; width: 2.562em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1002.5em, 4.229em, -999.997em); top: -6.009em; left: 50%; margin-left: -1.247em;"><span class="mtd" id="MathJax-Span-1395"><span class="mrow" id="MathJax-Span-1396"><span class="mo" id="MathJax-Span-1397" style="font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-1398" style="font-family: MathJax_Main;">0.58</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1001.73em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.89em;"><span class="mtd" id="MathJax-Span-1409"><span class="mrow" id="MathJax-Span-1410"><span class="mn" id="MathJax-Span-1411" style="font-family: MathJax_Main;">0.82</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1421"><span class="mrow" id="MathJax-Span-1422"><span class="mn" id="MathJax-Span-1423" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1433"><span class="mrow" id="MathJax-Span-1434"><span class="mn" id="MathJax-Span-1435" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1000.48em, 8.217em, -999.997em); top: -5.89em; left: 7.086em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1399"><span class="mrow" id="MathJax-Span-1400"><span class="mn" id="MathJax-Span-1401" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1412"><span class="mrow" id="MathJax-Span-1413"><span class="mn" id="MathJax-Span-1414" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1424"><span class="mrow" id="MathJax-Span-1425"><span class="mn" id="MathJax-Span-1426" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1436"><span class="mrow" id="MathJax-Span-1437"><span class="mn" id="MathJax-Span-1438" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1000.48em, 8.217em, -999.997em); top: -5.89em; left: 8.634em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1402"><span class="mrow" id="MathJax-Span-1403"><span class="mn" id="MathJax-Span-1404" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1415"><span class="mrow" id="MathJax-Span-1416"><span class="mn" id="MathJax-Span-1417" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1427"><span class="mrow" id="MathJax-Span-1428"><span class="mn" id="MathJax-Span-1429" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1439"><span class="mrow" id="MathJax-Span-1440"><span class="mn" id="MathJax-Span-1441" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span></span></span><span class="mo" id="MathJax-Span-1442" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎤<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.598em; left: 0em;">⎦<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.664em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.068em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.533em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span class="mo" id="MathJax-Span-1443" style="font-family: MathJax_Main; padding-left: 0.182em;">.</span></span><span style="display: inline-block; width: 0px; height: 5.479em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.854em; border-left: 0px solid; width: 0px; height: 6.361em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">U</mi></mrow><mo>=</mo><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mo>−</mo><mn>0.82</mn></mtd><mtd><mo>−</mo><mn>0.58</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>0.58</mn></mtd><mtd><mn>0.82</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable><mo>]</mo></mrow><mo>.</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-72">\mathbf U = \begin{bmatrix}-0.82 & -0.58 & 0 & 0 \\ -0.58 & 0.82 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}.</script><p class=" has-jax has-jax">同理可解得（注意，<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-73-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1444" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1002.74em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1445"><span class="texatom" id="MathJax-Span-1446"><span class="mrow" id="MathJax-Span-1447"><span class="mi" id="MathJax-Span-1448" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="msubsup" id="MathJax-Span-1449"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-1450"><span class="mrow" id="MathJax-Span-1451"><span class="mi" id="MathJax-Span-1452" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-1453"><span class="mrow" id="MathJax-Span-1454"><span class="texatom" id="MathJax-Span-1455"><span class="mrow" id="MathJax-Span-1456"><span class="mi" id="MathJax-Span-1457" style="font-size: 70.7%; font-family: MathJax_SansSerif;">T</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">T</mi></mrow></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-73">\mathbf M\mathbf M^{\mathsf T}</script> 和 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-74-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1458" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em, 1002.68em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1459"><span class="msubsup" id="MathJax-Span-1460"><span style="display: inline-block; position: relative; width: 1.67em; height: 0px;"><span style="position: absolute; clip: rect(3.098em, 1001.07em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-1461"><span class="mrow" id="MathJax-Span-1462"><span class="mi" id="MathJax-Span-1463" style="font-family: MathJax_Main-bold;">M</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 1.074em;"><span class="texatom" id="MathJax-Span-1464"><span class="mrow" id="MathJax-Span-1465"><span class="texatom" id="MathJax-Span-1466"><span class="mrow" id="MathJax-Span-1467"><span class="mi" id="MathJax-Span-1468" style="font-size: 70.7%; font-family: MathJax_SansSerif;">T</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="texatom" id="MathJax-Span-1469"><span class="mrow" id="MathJax-Span-1470"><span class="mi" id="MathJax-Span-1471" style="font-family: MathJax_Main-bold;">M</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="sans-serif">T</mi></mrow></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-74">\mathbf M^{\mathsf T}\mathbf M</script> 的特征值相同）</p>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-75-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;0.40&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;0.91&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;0.91&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0.40&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1472" style="width: 12.324em; display: inline-block;"><span style="display: inline-block; position: relative; width: 10.241em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.789em, 1010.18em, 4.527em, -999.997em); top: -3.39em; left: 0em;"><span class="mrow" id="MathJax-Span-1473"><span class="texatom" id="MathJax-Span-1474"><span class="mrow" id="MathJax-Span-1475"><span class="mi" id="MathJax-Span-1476" style="font-family: MathJax_Main-bold;">V</span></span></span><span class="mo" id="MathJax-Span-1477" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="mrow" id="MathJax-Span-1478" style="padding-left: 0.301em;"><span class="mo" id="MathJax-Span-1479" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mtable" id="MathJax-Span-1480" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 6.134em; height: 0px;"><span style="position: absolute; clip: rect(2.503em, 1002.5em, 5.003em, -999.997em); top: -3.985em; left: 0em;"><span style="display: inline-block; position: relative; width: 2.562em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1002.5em, 4.229em, -999.997em); top: -4.64em; left: 50%; margin-left: -1.247em;"><span class="mtd" id="MathJax-Span-1481"><span class="mrow" id="MathJax-Span-1482"><span class="mo" id="MathJax-Span-1483" style="font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-1484" style="font-family: MathJax_Main;">0.40</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1002.5em, 4.229em, -999.997em); top: -3.211em; left: 50%; margin-left: -1.247em;"><span class="mtd" id="MathJax-Span-1489"><span class="mrow" id="MathJax-Span-1490"><span class="mo" id="MathJax-Span-1491" style="font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-1492" style="font-family: MathJax_Main;">0.91</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(2.503em, 1002.5em, 4.943em, -999.997em); top: -3.985em; left: 3.574em;"><span style="display: inline-block; position: relative; width: 2.562em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1002.5em, 4.229em, -999.997em); top: -4.64em; left: 50%; margin-left: -1.247em;"><span class="mtd" id="MathJax-Span-1485"><span class="mrow" id="MathJax-Span-1486"><span class="mo" id="MathJax-Span-1487" style="font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-1488" style="font-family: MathJax_Main;">0.91</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1001.73em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.89em;"><span class="mtd" id="MathJax-Span-1493"><span class="mrow" id="MathJax-Span-1494"><span class="mn" id="MathJax-Span-1495" style="font-family: MathJax_Main;">0.40</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-1496" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">]</span></span></span><span class="mo" id="MathJax-Span-1497" style="font-family: MathJax_Main; padding-left: 0.182em;">.</span></span><span style="display: inline-block; width: 0px; height: 3.396em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.211em; border-left: 0px solid; width: 0px; height: 3.004em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">V</mi></mrow><mo>=</mo><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mo>−</mo><mn>0.40</mn></mtd><mtd><mo>−</mo><mn>0.91</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>0.91</mn></mtd><mtd><mn>0.40</mn></mtd></mtr></mtable><mo>]</mo></mrow><mo>.</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-75">\mathbf V = \begin{bmatrix}-0.40 & -0.91 \\ -0.91 & 0.40\end{bmatrix}.</script><p class=" has-jax has-jax">以及 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-76-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;&amp;#x03A3;&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1498" style="width: 1.015em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.836em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1000.78em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1499"><span class="texatom" id="MathJax-Span-1500"><span class="mrow" id="MathJax-Span-1501"><span class="mi" id="MathJax-Span-1502" style="font-family: MathJax_Main-bold;">Σ</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">Σ</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-76">\mathbf\Sigma</script> 上的对角线元素由 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-77-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;W&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1503" style="width: 1.432em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.193em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1001.19em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1504"><span class="texatom" id="MathJax-Span-1505"><span class="mrow" id="MathJax-Span-1506"><span class="mi" id="MathJax-Span-1507" style="font-family: MathJax_Main-bold;">W</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">W</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-77">\mathbf W</script> 的特征值的算术平方根组成；因此有</p>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-78-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;&amp;#x03A3;&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;5.46&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0.37&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1508" style="width: 10.717em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.932em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(2.443em, 1008.87em, 7.979em, -999.997em); top: -5.473em; left: 0em;"><span class="mrow" id="MathJax-Span-1509"><span class="texatom" id="MathJax-Span-1510"><span class="mrow" id="MathJax-Span-1511"><span class="mi" id="MathJax-Span-1512" style="font-family: MathJax_Main-bold;">Σ</span></span></span><span class="mo" id="MathJax-Span-1513" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="mrow" id="MathJax-Span-1514" style="padding-left: 0.301em;"><span class="mo" id="MathJax-Span-1515" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎡<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.598em; left: 0em;">⎣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.664em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.068em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.533em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mtable" id="MathJax-Span-1516" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 4.586em; height: 0px;"><span style="position: absolute; clip: rect(2.979em, 1001.73em, 8.217em, -999.997em); top: -5.89em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.789em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1001.73em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.89em;"><span class="mtd" id="MathJax-Span-1517"><span class="mrow" id="MathJax-Span-1518"><span class="mn" id="MathJax-Span-1519" style="font-family: MathJax_Main;">5.46</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1523"><span class="mrow" id="MathJax-Span-1524"><span class="mn" id="MathJax-Span-1525" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1529"><span class="mrow" id="MathJax-Span-1530"><span class="mn" id="MathJax-Span-1531" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1535"><span class="mrow" id="MathJax-Span-1536"><span class="mn" id="MathJax-Span-1537" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1001.79em, 8.217em, -999.997em); top: -5.89em; left: 2.801em;"><span style="display: inline-block; position: relative; width: 1.789em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1520"><span class="mrow" id="MathJax-Span-1521"><span class="mn" id="MathJax-Span-1522" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1001.79em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.89em;"><span class="mtd" id="MathJax-Span-1526"><span class="mrow" id="MathJax-Span-1527"><span class="mn" id="MathJax-Span-1528" style="font-family: MathJax_Main;">0.37</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1532"><span class="mrow" id="MathJax-Span-1533"><span class="mn" id="MathJax-Span-1534" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1538"><span class="mrow" id="MathJax-Span-1539"><span class="mn" id="MathJax-Span-1540" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span></span></span><span class="mo" id="MathJax-Span-1541" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎤<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.598em; left: 0em;">⎦<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.664em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.068em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.533em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span class="mo" id="MathJax-Span-1542" style="font-family: MathJax_Main; padding-left: 0.182em;">.</span></span><span style="display: inline-block; width: 0px; height: 5.479em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.854em; border-left: 0px solid; width: 0px; height: 6.361em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">Σ</mi></mrow><mo>=</mo><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>5.46</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0.37</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo>]</mo></mrow><mo>.</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-78">\mathbf\Sigma = \begin{bmatrix}5.46 & 0 \\ 0 & 0.37 \\ 0 & 0 \\ 0 & 0\end{bmatrix}.</script><p class=" has-jax">因此我们得到矩阵 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-79-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1543" style="width: 1.313em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.074em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1001.01em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1544"><span class="texatom" id="MathJax-Span-1545"><span class="mrow" id="MathJax-Span-1546"><span class="mi" id="MathJax-Span-1547" style="font-family: MathJax_Main-bold;">M</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-79">\mathbf M</script> 的 SVD 分解（数值上做了近似）：</p>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-80-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2248;&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;0.82&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;0.58&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;0.58&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0.82&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;5.46&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0.37&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;0.40&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;0.91&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;0.91&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0.40&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1548" style="width: 36.074em; display: inline-block;"><span style="display: inline-block; position: relative; width: 30.063em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(2.979em, 1029.82em, 8.515em, -999.997em); top: -6.009em; left: 0em;"><span class="mrow" id="MathJax-Span-1549"><span class="mrow" id="MathJax-Span-1550"><span class="mo" id="MathJax-Span-1551" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎡<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.598em; left: 0em;">⎣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.664em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.068em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.533em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mtable" id="MathJax-Span-1552" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 2.027em; height: 0px;"><span style="position: absolute; clip: rect(2.979em, 1000.48em, 8.217em, -999.997em); top: -5.89em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1553"><span class="mrow" id="MathJax-Span-1554"><span class="mn" id="MathJax-Span-1555" style="font-family: MathJax_Main;">2</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1559"><span class="mrow" id="MathJax-Span-1560"><span class="mn" id="MathJax-Span-1561" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1565"><span class="mrow" id="MathJax-Span-1566"><span class="mn" id="MathJax-Span-1567" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1571"><span class="mrow" id="MathJax-Span-1572"><span class="mn" id="MathJax-Span-1573" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1000.48em, 8.217em, -999.997em); top: -5.89em; left: 1.491em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1556"><span class="mrow" id="MathJax-Span-1557"><span class="mn" id="MathJax-Span-1558" style="font-family: MathJax_Main;">4</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1562"><span class="mrow" id="MathJax-Span-1563"><span class="mn" id="MathJax-Span-1564" style="font-family: MathJax_Main;">3</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1568"><span class="mrow" id="MathJax-Span-1569"><span class="mn" id="MathJax-Span-1570" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1574"><span class="mrow" id="MathJax-Span-1575"><span class="mn" id="MathJax-Span-1576" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span></span></span><span class="mo" id="MathJax-Span-1577" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎤<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.598em; left: 0em;">⎦<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.664em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.068em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.533em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span class="mo" id="MathJax-Span-1578" style="font-family: MathJax_Main; padding-left: 0.301em;">≈</span><span class="mrow" id="MathJax-Span-1579" style="padding-left: 0.301em;"><span class="mo" id="MathJax-Span-1580" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎡<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.598em; left: 0em;">⎣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.664em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.068em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.533em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mtable" id="MathJax-Span-1581" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 9.11em; height: 0px;"><span style="position: absolute; clip: rect(2.979em, 1002.5em, 8.217em, -999.997em); top: -5.89em; left: 0em;"><span style="display: inline-block; position: relative; width: 2.562em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1002.5em, 4.229em, -999.997em); top: -6.009em; left: 50%; margin-left: -1.247em;"><span class="mtd" id="MathJax-Span-1582"><span class="mrow" id="MathJax-Span-1583"><span class="mo" id="MathJax-Span-1584" style="font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-1585" style="font-family: MathJax_Main;">0.82</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1002.5em, 4.229em, -999.997em); top: -4.64em; left: 50%; margin-left: -1.247em;"><span class="mtd" id="MathJax-Span-1596"><span class="mrow" id="MathJax-Span-1597"><span class="mo" id="MathJax-Span-1598" style="font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-1599" style="font-family: MathJax_Main;">0.58</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1609"><span class="mrow" id="MathJax-Span-1610"><span class="mn" id="MathJax-Span-1611" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1621"><span class="mrow" id="MathJax-Span-1622"><span class="mn" id="MathJax-Span-1623" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1002.5em, 8.217em, -999.997em); top: -5.89em; left: 3.574em;"><span style="display: inline-block; position: relative; width: 2.562em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1002.5em, 4.229em, -999.997em); top: -6.009em; left: 50%; margin-left: -1.247em;"><span class="mtd" id="MathJax-Span-1586"><span class="mrow" id="MathJax-Span-1587"><span class="mo" id="MathJax-Span-1588" style="font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-1589" style="font-family: MathJax_Main;">0.58</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1001.73em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.89em;"><span class="mtd" id="MathJax-Span-1600"><span class="mrow" id="MathJax-Span-1601"><span class="mn" id="MathJax-Span-1602" style="font-family: MathJax_Main;">0.82</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1612"><span class="mrow" id="MathJax-Span-1613"><span class="mn" id="MathJax-Span-1614" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1624"><span class="mrow" id="MathJax-Span-1625"><span class="mn" id="MathJax-Span-1626" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1000.48em, 8.217em, -999.997em); top: -5.89em; left: 7.086em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1590"><span class="mrow" id="MathJax-Span-1591"><span class="mn" id="MathJax-Span-1592" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1603"><span class="mrow" id="MathJax-Span-1604"><span class="mn" id="MathJax-Span-1605" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1615"><span class="mrow" id="MathJax-Span-1616"><span class="mn" id="MathJax-Span-1617" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1627"><span class="mrow" id="MathJax-Span-1628"><span class="mn" id="MathJax-Span-1629" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1000.48em, 8.217em, -999.997em); top: -5.89em; left: 8.634em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1593"><span class="mrow" id="MathJax-Span-1594"><span class="mn" id="MathJax-Span-1595" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1606"><span class="mrow" id="MathJax-Span-1607"><span class="mn" id="MathJax-Span-1608" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1618"><span class="mrow" id="MathJax-Span-1619"><span class="mn" id="MathJax-Span-1620" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1630"><span class="mrow" id="MathJax-Span-1631"><span class="mn" id="MathJax-Span-1632" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span></span></span><span class="mo" id="MathJax-Span-1633" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎤<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.598em; left: 0em;">⎦<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.664em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.068em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.533em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span class="mrow" id="MathJax-Span-1634" style="padding-left: 0.182em;"><span class="mo" id="MathJax-Span-1635" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎡<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.598em; left: 0em;">⎣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.664em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.068em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.533em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mtable" id="MathJax-Span-1636" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 4.586em; height: 0px;"><span style="position: absolute; clip: rect(2.979em, 1001.73em, 8.217em, -999.997em); top: -5.89em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.789em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1001.73em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.89em;"><span class="mtd" id="MathJax-Span-1637"><span class="mrow" id="MathJax-Span-1638"><span class="mn" id="MathJax-Span-1639" style="font-family: MathJax_Main;">5.46</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1643"><span class="mrow" id="MathJax-Span-1644"><span class="mn" id="MathJax-Span-1645" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1649"><span class="mrow" id="MathJax-Span-1650"><span class="mn" id="MathJax-Span-1651" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1655"><span class="mrow" id="MathJax-Span-1656"><span class="mn" id="MathJax-Span-1657" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1001.79em, 8.217em, -999.997em); top: -5.89em; left: 2.801em;"><span style="display: inline-block; position: relative; width: 1.789em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1640"><span class="mrow" id="MathJax-Span-1641"><span class="mn" id="MathJax-Span-1642" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1001.79em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.89em;"><span class="mtd" id="MathJax-Span-1646"><span class="mrow" id="MathJax-Span-1647"><span class="mn" id="MathJax-Span-1648" style="font-family: MathJax_Main;">0.37</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1652"><span class="mrow" id="MathJax-Span-1653"><span class="mn" id="MathJax-Span-1654" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1658"><span class="mrow" id="MathJax-Span-1659"><span class="mn" id="MathJax-Span-1660" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span></span></span><span class="mo" id="MathJax-Span-1661" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.658em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.854em; left: 0em;">⎤<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.598em; left: 0em;">⎦<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.664em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.068em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.533em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span class="mrow" id="MathJax-Span-1662" style="padding-left: 0.182em;"><span class="mo" id="MathJax-Span-1663" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">[</span></span><span class="mtable" id="MathJax-Span-1664" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 6.134em; height: 0px;"><span style="position: absolute; clip: rect(2.503em, 1002.5em, 5.003em, -999.997em); top: -3.985em; left: 0em;"><span style="display: inline-block; position: relative; width: 2.562em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1002.5em, 4.229em, -999.997em); top: -4.64em; left: 50%; margin-left: -1.247em;"><span class="mtd" id="MathJax-Span-1665"><span class="mrow" id="MathJax-Span-1666"><span class="mo" id="MathJax-Span-1667" style="font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-1668" style="font-family: MathJax_Main;">0.40</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1002.5em, 4.229em, -999.997em); top: -3.211em; left: 50%; margin-left: -1.247em;"><span class="mtd" id="MathJax-Span-1673"><span class="mrow" id="MathJax-Span-1674"><span class="mo" id="MathJax-Span-1675" style="font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-1676" style="font-family: MathJax_Main;">0.91</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(2.503em, 1002.5em, 4.943em, -999.997em); top: -3.985em; left: 3.574em;"><span style="display: inline-block; position: relative; width: 2.562em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1002.5em, 4.229em, -999.997em); top: -4.64em; left: 50%; margin-left: -1.247em;"><span class="mtd" id="MathJax-Span-1669"><span class="mrow" id="MathJax-Span-1670"><span class="mo" id="MathJax-Span-1671" style="font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-1672" style="font-family: MathJax_Main;">0.91</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1001.73em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.89em;"><span class="mtd" id="MathJax-Span-1677"><span class="mrow" id="MathJax-Span-1678"><span class="mn" id="MathJax-Span-1679" style="font-family: MathJax_Main;">0.40</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-1680" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">]</span></span></span></span><span style="display: inline-block; width: 0px; height: 6.015em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.854em; border-left: 0px solid; width: 0px; height: 6.361em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>2</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo>]</mo></mrow><mo>≈</mo><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mo>−</mo><mn>0.82</mn></mtd><mtd><mo>−</mo><mn>0.58</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>0.58</mn></mtd><mtd><mn>0.82</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable><mo>]</mo></mrow><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>5.46</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0.37</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo>]</mo></mrow><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mo>−</mo><mn>0.40</mn></mtd><mtd><mo>−</mo><mn>0.91</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>0.91</mn></mtd><mtd><mn>0.40</mn></mtd></mtr></mtable><mo>]</mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-80">\begin{bmatrix}2 & 4 \\ 1 & 3 \\ 0 & 0 \\ 0 & 0\end{bmatrix} \approx \begin{bmatrix}-0.82 & -0.58 & 0 & 0 \\ -0.58 & 0.82 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}5.46 & 0 \\ 0 & 0.37 \\ 0 & 0 \\ 0 & 0\end{bmatrix}\begin{bmatrix}-0.40 & -0.91 \\ -0.91 & 0.40\end{bmatrix}</script><h2 id="几何上的直观解释"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E5%87%A0%E4%BD%95%E4%B8%8A%E7%9A%84%E7%9B%B4%E8%A7%82%E8%A7%A3%E9%87%8A" class="headerlink" title="几何上的直观解释"></a>几何上的直观解释</h2><p class=" has-jax has-jax has-jax has-jax has-jax has-jax has-jax has-jax has-jax has-jax has-jax">我们先来看一个例子。假设 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-81-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1681" style="width: 1.313em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.074em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1001.01em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1682"><span class="texatom" id="MathJax-Span-1683"><span class="mrow" id="MathJax-Span-1684"><span class="mi" id="MathJax-Span-1685" style="font-family: MathJax_Main-bold;">M</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-81">\mathbf M</script> 是一个 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-82-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1686" style="width: 3.336em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.741em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.967em, 1002.74em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1687"><span class="mi" id="MathJax-Span-1688" style="font-family: MathJax_Math-italic;">m</span><span class="mo" id="MathJax-Span-1689" style="font-family: MathJax_Main; padding-left: 0.241em;">×</span><span class="mi" id="MathJax-Span-1690" style="font-family: MathJax_Math-italic; padding-left: 0.241em;">n</span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 0.718em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>×</mo><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-82">m\times n</script> 的矩阵，而 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-83-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1691" style="width: 0.717em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.598em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(2.027em, 1000.6em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1692"><span class="texatom" id="MathJax-Span-1693"><span class="mrow" id="MathJax-Span-1694"><span class="mi" id="MathJax-Span-1695" style="font-family: MathJax_Main-bold;">x</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 0.646em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-83">\mathbf x</script> 是线性空间 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-84-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1696" style="width: 1.551em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.253em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1001.25em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1697"><span class="msubsup" id="MathJax-Span-1698"><span style="display: inline-block; position: relative; width: 1.253em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.78em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-1699"><span class="mrow" id="MathJax-Span-1700"><span class="mi" id="MathJax-Span-1701" style="font-family: MathJax_AMS;">K</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.777em;"><span class="mi" id="MathJax-Span-1702" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">K</mi></mrow><mi>n</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-84">\mathbb K^n</script> 中的向量，则<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-85-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x22C5;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1703" style="width: 5.36em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.467em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1004.47em, 2.979em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1704"><span class="texatom" id="MathJax-Span-1705"><span class="mrow" id="MathJax-Span-1706"><span class="mi" id="MathJax-Span-1707" style="font-family: MathJax_Main-bold;">y</span></span></span><span class="mo" id="MathJax-Span-1708" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="texatom" id="MathJax-Span-1709" style="padding-left: 0.301em;"><span class="mrow" id="MathJax-Span-1710"><span class="mi" id="MathJax-Span-1711" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="mo" id="MathJax-Span-1712" style="font-family: MathJax_Main; padding-left: 0.241em;">⋅</span><span class="texatom" id="MathJax-Span-1713" style="padding-left: 0.241em;"><span class="mrow" id="MathJax-Span-1714"><span class="mi" id="MathJax-Span-1715" style="font-family: MathJax_Main-bold;">x</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">y</mi></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-85">\mathbf y = \mathbf M\cdot\mathbf x</script> 是线性空间 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-86-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1716" style="width: 1.789em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.491em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1001.49em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1717"><span class="msubsup" id="MathJax-Span-1718"><span style="display: inline-block; position: relative; width: 1.491em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.78em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-1719"><span class="mrow" id="MathJax-Span-1720"><span class="mi" id="MathJax-Span-1721" style="font-family: MathJax_AMS;">K</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.777em;"><span class="mi" id="MathJax-Span-1722" style="font-size: 70.7%; font-family: MathJax_Math-italic;">m</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">K</mi></mrow><mi>m</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-86">\mathbb K^m</script> 中的向量。这样一来，矩阵 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-87-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1723" style="width: 0.896em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.717em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.789em, 1000.72em, 2.86em, -999.997em); top: -2.676em; left: 0em;"><span class="mrow" id="MathJax-Span-1724"><span class="texatom" id="MathJax-Span-1725"><span class="mrow" id="MathJax-Span-1726"><span class="mi" id="MathJax-Span-1727" style="font-family: MathJax_AMS;">A</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.682em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">A</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-87">\mathbb A</script> 就对应了一个从 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-88-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1728" style="width: 1.551em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.253em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1001.25em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1729"><span class="msubsup" id="MathJax-Span-1730"><span style="display: inline-block; position: relative; width: 1.253em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.78em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-1731"><span class="mrow" id="MathJax-Span-1732"><span class="mi" id="MathJax-Span-1733" style="font-family: MathJax_AMS;">K</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.777em;"><span class="mi" id="MathJax-Span-1734" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">K</mi></mrow><mi>n</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-88">\mathbb K^n</script> 到 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-89-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1735" style="width: 1.789em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.491em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1001.49em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1736"><span class="msubsup" id="MathJax-Span-1737"><span style="display: inline-block; position: relative; width: 1.491em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.78em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-1738"><span class="mrow" id="MathJax-Span-1739"><span class="mi" id="MathJax-Span-1740" style="font-family: MathJax_AMS;">K</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.777em;"><span class="mi" id="MathJax-Span-1741" style="font-size: 70.7%; font-family: MathJax_Math-italic;">m</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">K</mi></mrow><mi>m</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-89">\mathbb K^m</script> 的变换 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-90-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x2192;&lt;/mo&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1742" style="width: 7.146em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.955em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1005.96em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1743"><span class="mi" id="MathJax-Span-1744" style="font-family: MathJax_Math-italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.122em;"></span></span><span class="mo" id="MathJax-Span-1745" style="font-family: MathJax_Main; padding-left: 0.301em;">:</span><span class="msubsup" id="MathJax-Span-1746" style="padding-left: 0.301em;"><span style="display: inline-block; position: relative; width: 1.253em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.78em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-1747"><span class="mrow" id="MathJax-Span-1748"><span class="mi" id="MathJax-Span-1749" style="font-family: MathJax_AMS;">K</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.777em;"><span class="mi" id="MathJax-Span-1750" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-1751" style="font-family: MathJax_Main; padding-left: 0.301em;">→</span><span class="msubsup" id="MathJax-Span-1752" style="padding-left: 0.301em;"><span style="display: inline-block; position: relative; width: 1.491em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.78em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="texatom" id="MathJax-Span-1753"><span class="mrow" id="MathJax-Span-1754"><span class="mi" id="MathJax-Span-1755" style="font-family: MathJax_AMS;">K</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.402em; left: 0.777em;"><span class="mi" id="MathJax-Span-1756" style="font-size: 70.7%; font-family: MathJax_Math-italic;">m</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>:</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">K</mi></mrow><mi>n</mi></msup><mo stretchy="false">→</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">K</mi></mrow><mi>m</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-90">T: \mathbb K^n \to \mathbb K^m</script>，具体来说既是 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-91-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x21A6;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x22C5;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;bold&quot;&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1757" style="width: 5.598em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.646em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em, 1004.65em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1758"><span class="texatom" id="MathJax-Span-1759"><span class="mrow" id="MathJax-Span-1760"><span class="mi" id="MathJax-Span-1761" style="font-family: MathJax_Main-bold;">x</span></span></span><span class="mo" id="MathJax-Span-1762" style="font-family: MathJax_Main; padding-left: 0.301em;">↦</span><span class="texatom" id="MathJax-Span-1763" style="padding-left: 0.301em;"><span class="mrow" id="MathJax-Span-1764"><span class="mi" id="MathJax-Span-1765" style="font-family: MathJax_Main-bold;">M</span></span></span><span class="mo" id="MathJax-Span-1766" style="font-family: MathJax_Main; padding-left: 0.241em;">⋅</span><span class="texatom" id="MathJax-Span-1767" style="padding-left: 0.241em;"><span class="mrow" id="MathJax-Span-1768"><span class="mi" id="MathJax-Span-1769" style="font-family: MathJax_Main-bold;">x</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow><mo stretchy="false">↦</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">M</mi></mrow><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-91">\mathbf x\mapsto \mathbf M\cdot\mathbf x</script>。这也就是说，在线性代数中，任意矩阵都能看做是一种变换。这样一来，我们就统一了矩阵和变换。</p>
<h2 id="SVD-场景"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#SVD-%E5%9C%BA%E6%99%AF" class="headerlink" title="SVD 场景"></a>SVD 场景</h2><blockquote>
<p>隐性语义检索</p>
</blockquote>
<p>信息检索-隐性语义检索（Lstent Semantic Indexing, LSI）或 隐形语义分析（Latent Semantic Analysis, LSA）<br>隐性语义索引：矩阵 = 文档 + 词语<br>最早的 SVD 应用之一，我们称利用 SVD 的方法为隐性语义索引（LSI）或隐性语义分析（LSA）。</p>
<p><img src="https://i.imgur.com/EPoEEET.png" alt="" data-bd-imgshare-binded="1"></p>
<blockquote>
<p>推荐系统</p>
</blockquote>
<ol>
<li>利用 SVD 从数据中构建一个主题空间。</li>
<li>再在该空间下计算其相似度。(从高维-低维空间的转化，在低维空间来计算相似度，SVD 提升了推荐系统的效率。)</li>
</ol>
<blockquote>
<p>图像压缩</p>
</blockquote>
<p>例如：<code>32*32=1024 =&gt; 32*2+2*1+32*2=130</code>(2*1表示去掉了除对角线的0), 几乎获得了10倍的压缩比。</p>
<p><img src="https://i.imgur.com/yb6uATG.jpg" alt="" data-bd-imgshare-binded="1"></p>
<hr>
<h1 id="SVD-工作原理"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#SVD-%E5%B7%A5%E4%BD%9C%E5%8E%9F%E7%90%86" class="headerlink" title="SVD 工作原理"></a>SVD 工作原理</h1><h2 id="矩阵分解"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E7%9F%A9%E9%98%B5%E5%88%86%E8%A7%A3" class="headerlink" title="矩阵分解"></a>矩阵分解</h2><ul>
<li>矩阵分解是将数据矩阵分解为多个独立部分的过程。</li>
<li>矩阵分解可以将原始矩阵表示成新的易于处理的形式，这种新形式是两个或多个矩阵的乘积。（类似代数中的因数分解）</li>
<li>举例：如何将12分解成两个数的乘积？（1，12）、（2，6）、（3，4）都是合理的答案。</li>
</ul>
<h2 id="SVD-是矩阵分解的一种类型，也是矩阵分解最常见的技术"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#SVD-%E6%98%AF%E7%9F%A9%E9%98%B5%E5%88%86%E8%A7%A3%E7%9A%84%E4%B8%80%E7%A7%8D%E7%B1%BB%E5%9E%8B%EF%BC%8C%E4%B9%9F%E6%98%AF%E7%9F%A9%E9%98%B5%E5%88%86%E8%A7%A3%E6%9C%80%E5%B8%B8%E8%A7%81%E7%9A%84%E6%8A%80%E6%9C%AF" class="headerlink" title="SVD 是矩阵分解的一种类型，也是矩阵分解最常见的技术"></a>SVD 是矩阵分解的一种类型，也是矩阵分解最常见的技术</h2><ul>
<li>SVD 将原始的数据集矩阵 Data 分解成三个矩阵 U、∑、V</li>
<li class=" has-jax">举例：如果原始矩阵 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-92-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;&amp;#x2217;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1770" style="width: 4.527em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.753em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.789em, 1003.75em, 2.979em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1771"><span class="mi" id="MathJax-Span-1772" style="font-family: MathJax_Math-italic;">D</span><span class="mi" id="MathJax-Span-1773" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-1774" style="font-family: MathJax_Math-italic;">t</span><span class="msubsup" id="MathJax-Span-1775"><span style="display: inline-block; position: relative; width: 2.027em; height: 0px;"><span style="position: absolute; clip: rect(3.396em, 1000.54em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-1776" style="font-family: MathJax_Math-italic;">a</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.807em; left: 0.539em;"><span class="texatom" id="MathJax-Span-1777"><span class="mrow" id="MathJax-Span-1778"><span class="mi" id="MathJax-Span-1779" style="font-size: 70.7%; font-family: MathJax_Math-italic;">m</span><span class="mo" id="MathJax-Span-1780" style="font-size: 70.7%; font-family: MathJax_Main;">∗</span><span class="mi" id="MathJax-Span-1781" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.146em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mi>a</mi><mi>t</mi><msub><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mo>∗</mo><mi>n</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-92">Data_{m*n} </script> 是m行n列，<ul>
<li class=" has-jax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-93-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;&amp;#x2217;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1782" style="width: 2.503em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.086em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.789em, 1002.09em, 2.979em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1783"><span class="msubsup" id="MathJax-Span-1784"><span style="display: inline-block; position: relative; width: 2.086em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.78em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-1785" style="font-family: MathJax_Math-italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.807em; left: 0.658em;"><span class="texatom" id="MathJax-Span-1786"><span class="mrow" id="MathJax-Span-1787"><span class="mi" id="MathJax-Span-1788" style="font-size: 70.7%; font-family: MathJax_Math-italic;">m</span><span class="mo" id="MathJax-Span-1789" style="font-size: 70.7%; font-family: MathJax_Main;">∗</span><span class="mi" id="MathJax-Span-1790" style="font-size: 70.7%; font-family: MathJax_Math-italic;">k</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.146em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mo>∗</mo><mi>k</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-93">U_{m * k}</script> 表示m行k列</li>
<li class=" has-jax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-94-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2217;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1791" style="width: 2.682em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.205em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.67em, 1002.21em, 3.098em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1792"><span class="msubsup" id="MathJax-Span-1793"><span style="display: inline-block; position: relative; width: 2.205em; height: 0px;"><span style="position: absolute; clip: rect(3.039em, 1001.01em, 4.408em, -999.997em); top: -3.985em; left: 0em;"><span class="mo" id="MathJax-Span-1794" style="font-family: MathJax_Size1; vertical-align: 0em;">∑</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.688em; left: 1.074em;"><span class="texatom" id="MathJax-Span-1795"><span class="mrow" id="MathJax-Span-1796"><span class="mi" id="MathJax-Span-1797" style="font-size: 70.7%; font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-1798" style="font-size: 70.7%; font-family: MathJax_Main;">∗</span><span class="mi" id="MathJax-Span-1799" style="font-size: 70.7%; font-family: MathJax_Math-italic;">k</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.425em; border-left: 0px solid; width: 0px; height: 1.432em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>∗</mo><mi>k</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-94">∑_{k * k}</script> 表示k行k列</li>
<li class=" has-jax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-95-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#x2217;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1800" style="width: 2.146em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.789em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.789em, 1001.79em, 2.979em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1801"><span class="msubsup" id="MathJax-Span-1802"><span style="display: inline-block; position: relative; width: 1.789em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.78em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-1803" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.182em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.807em; left: 0.598em;"><span class="texatom" id="MathJax-Span-1804"><span class="mrow" id="MathJax-Span-1805"><span class="mi" id="MathJax-Span-1806" style="font-size: 70.7%; font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-1807" style="font-size: 70.7%; font-family: MathJax_Main;">∗</span><span class="mi" id="MathJax-Span-1808" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.146em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>∗</mo><mi>n</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-95">V_{k * n}</script> 表示k行n列。</li>
</ul>
</li>
</ul>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-96-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;&amp;#xD7;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;&amp;#xD7;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2217;&lt;/mo&gt;&lt;msub&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#xD7;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2217;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&amp;#xD7;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1809" style="width: 16.967em; display: inline-block;"><span style="display: inline-block; position: relative; width: 14.11em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.491em, 1014.11em, 3.277em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-1810"><span class="mi" id="MathJax-Span-1811" style="font-family: MathJax_Math-italic;">D</span><span class="mi" id="MathJax-Span-1812" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-1813" style="font-family: MathJax_Math-italic;">t</span><span class="msubsup" id="MathJax-Span-1814"><span style="display: inline-block; position: relative; width: 2.205em; height: 0px;"><span style="position: absolute; clip: rect(3.396em, 1000.54em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-1815" style="font-family: MathJax_Math-italic;">a</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.807em; left: 0.539em;"><span class="texatom" id="MathJax-Span-1816"><span class="mrow" id="MathJax-Span-1817"><span class="mi" id="MathJax-Span-1818" style="font-size: 70.7%; font-family: MathJax_Math-italic;">m</span><span class="mo" id="MathJax-Span-1819" style="font-size: 70.7%; font-family: MathJax_Main;">×</span><span class="mi" id="MathJax-Span-1820" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-1821" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="msubsup" id="MathJax-Span-1822" style="padding-left: 0.301em;"><span style="display: inline-block; position: relative; width: 2.324em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.78em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-1823" style="font-family: MathJax_Math-italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.807em; left: 0.658em;"><span class="texatom" id="MathJax-Span-1824"><span class="mrow" id="MathJax-Span-1825"><span class="mi" id="MathJax-Span-1826" style="font-size: 70.7%; font-family: MathJax_Math-italic;">m</span><span class="mo" id="MathJax-Span-1827" style="font-size: 70.7%; font-family: MathJax_Main;">×</span><span class="mi" id="MathJax-Span-1828" style="font-size: 70.7%; font-family: MathJax_Math-italic;">k</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-1829" style="font-family: MathJax_Main; padding-left: 0.241em;">∗</span><span class="msubsup" id="MathJax-Span-1830" style="padding-left: 0.241em;"><span style="display: inline-block; position: relative; width: 2.801em; height: 0px;"><span style="position: absolute; clip: rect(2.86em, 1001.37em, 4.646em, -999.997em); top: -3.985em; left: 0em;"><span class="mo" id="MathJax-Span-1831" style="font-family: MathJax_Size2; vertical-align: 0em;">∑</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.509em; left: 1.432em;"><span class="texatom" id="MathJax-Span-1832"><span class="mrow" id="MathJax-Span-1833"><span class="mi" id="MathJax-Span-1834" style="font-size: 70.7%; font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-1835" style="font-size: 70.7%; font-family: MathJax_Main;">×</span><span class="mi" id="MathJax-Span-1836" style="font-size: 70.7%; font-family: MathJax_Math-italic;">k</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-1837" style="font-family: MathJax_Main; padding-left: 0.182em;">∗</span><span class="msubsup" id="MathJax-Span-1838"><span style="display: inline-block; position: relative; width: 2.027em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.78em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-1839" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.182em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.807em; left: 0.598em;"><span class="texatom" id="MathJax-Span-1840"><span class="mrow" id="MathJax-Span-1841"><span class="mi" id="MathJax-Span-1842" style="font-size: 70.7%; font-family: MathJax_Math-italic;">k</span><span class="mo" id="MathJax-Span-1843" style="font-size: 70.7%; font-family: MathJax_Main;">×</span><span class="mi" id="MathJax-Span-1844" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.639em; border-left: 0px solid; width: 0px; height: 1.861em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>D</mi><mi>a</mi><mi>t</mi><msub><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msub><mo>=</mo><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mo>×</mo><mi>k</mi></mrow></msub><mo>∗</mo><msub><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>×</mo><mi>k</mi></mrow></msub><mo>∗</mo><msub><mi>V</mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>×</mo><mi>n</mi></mrow></msub></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-96">Data_{m×n} = U_{m×k} * ∑_{k×k} * V_{k×n}</script><p><img src="https://i.imgur.com/fbVpRi1.jpg" alt="" data-bd-imgshare-binded="1"></p>
<p>具体的案例：</p>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-97-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1.6&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0.6&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1.2&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0.8&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0.8&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0.6&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;0.6&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0.8&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2217;&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2217;&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1845" style="width: 33.812em; display: inline-block;"><span style="display: inline-block; position: relative; width: 28.158em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(2.443em, 1028.04em, 7.979em, -999.997em); top: -5.473em; left: 0em;"><span class="mrow" id="MathJax-Span-1846"><span class="mrow" id="MathJax-Span-1847"><span class="mo" id="MathJax-Span-1848" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.301em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.211em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: 0.955em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -2.378em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -1.545em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -0.711em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: 0.122em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mtable" id="MathJax-Span-1849" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 5.836em; height: 0px;"><span style="position: absolute; clip: rect(2.979em, 1000.48em, 8.217em, -999.997em); top: -5.89em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1850"><span class="mrow" id="MathJax-Span-1851"><span class="mn" id="MathJax-Span-1852" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1860"><span class="mrow" id="MathJax-Span-1861"><span class="mn" id="MathJax-Span-1862" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1869"><span class="mrow" id="MathJax-Span-1870"><span class="mn" id="MathJax-Span-1871" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1878"><span class="mrow" id="MathJax-Span-1879"><span class="mn" id="MathJax-Span-1880" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1002.03em, 8.217em, -999.997em); top: -5.89em; left: 1.491em;"><span style="display: inline-block; position: relative; width: 2.086em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1002.03em, 4.229em, -999.997em); top: -6.009em; left: 50%; margin-left: -1.009em;"><span class="mtd" id="MathJax-Span-1853"><span class="mrow" id="MathJax-Span-1854"><span class="mo" id="MathJax-Span-1855" style="font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-1856" style="font-family: MathJax_Main;">1.6</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1001.25em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.652em;"><span class="mtd" id="MathJax-Span-1863"><span class="mrow" id="MathJax-Span-1864"><span class="mn" id="MathJax-Span-1865" style="font-family: MathJax_Main;">1.2</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1872"><span class="mrow" id="MathJax-Span-1873"><span class="mn" id="MathJax-Span-1874" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1881"><span class="mrow" id="MathJax-Span-1882"><span class="mn" id="MathJax-Span-1883" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1001.25em, 8.217em, -999.997em); top: -5.89em; left: 4.586em;"><span style="display: inline-block; position: relative; width: 1.253em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1001.25em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.652em;"><span class="mtd" id="MathJax-Span-1857"><span class="mrow" id="MathJax-Span-1858"><span class="mn" id="MathJax-Span-1859" style="font-family: MathJax_Main;">0.6</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1001.25em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.652em;"><span class="mtd" id="MathJax-Span-1866"><span class="mrow" id="MathJax-Span-1867"><span class="mn" id="MathJax-Span-1868" style="font-family: MathJax_Main;">0.8</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1875"><span class="mrow" id="MathJax-Span-1876"><span class="mn" id="MathJax-Span-1877" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1884"><span class="mrow" id="MathJax-Span-1885"><span class="mn" id="MathJax-Span-1886" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span></span></span><span class="mo" id="MathJax-Span-1887" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.301em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.211em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: 0.955em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -2.378em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -1.545em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -0.711em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: 0.122em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span class="mo" id="MathJax-Span-1888" style="font-family: MathJax_Main; padding-left: 0.301em;">=</span><span class="mrow" id="MathJax-Span-1889" style="padding-left: 0.301em;"><span class="mo" id="MathJax-Span-1890" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.301em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.211em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: 0.955em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -2.378em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -1.545em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -0.711em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: 0.122em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mtable" id="MathJax-Span-1891" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 7.324em; height: 0px;"><span style="position: absolute; clip: rect(2.979em, 1002.03em, 8.217em, -999.997em); top: -5.89em; left: 0em;"><span style="display: inline-block; position: relative; width: 2.086em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1001.25em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.652em;"><span class="mtd" id="MathJax-Span-1892"><span class="mrow" id="MathJax-Span-1893"><span class="mn" id="MathJax-Span-1894" style="font-family: MathJax_Main;">0.8</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1002.03em, 4.229em, -999.997em); top: -4.64em; left: 50%; margin-left: -1.009em;"><span class="mtd" id="MathJax-Span-1904"><span class="mrow" id="MathJax-Span-1905"><span class="mo" id="MathJax-Span-1906" style="font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-1907" style="font-family: MathJax_Main;">0.6</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1917"><span class="mrow" id="MathJax-Span-1918"><span class="mn" id="MathJax-Span-1919" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1929"><span class="mrow" id="MathJax-Span-1930"><span class="mn" id="MathJax-Span-1931" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1001.25em, 8.217em, -999.997em); top: -5.89em; left: 3.039em;"><span style="display: inline-block; position: relative; width: 1.253em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1001.25em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.652em;"><span class="mtd" id="MathJax-Span-1895"><span class="mrow" id="MathJax-Span-1896"><span class="mn" id="MathJax-Span-1897" style="font-family: MathJax_Main;">0.6</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1001.25em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.652em;"><span class="mtd" id="MathJax-Span-1908"><span class="mrow" id="MathJax-Span-1909"><span class="mn" id="MathJax-Span-1910" style="font-family: MathJax_Main;">0.8</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1920"><span class="mrow" id="MathJax-Span-1921"><span class="mn" id="MathJax-Span-1922" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1932"><span class="mrow" id="MathJax-Span-1933"><span class="mn" id="MathJax-Span-1934" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1000.48em, 8.217em, -999.997em); top: -5.89em; left: 5.36em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1898"><span class="mrow" id="MathJax-Span-1899"><span class="mn" id="MathJax-Span-1900" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1911"><span class="mrow" id="MathJax-Span-1912"><span class="mn" id="MathJax-Span-1913" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1923"><span class="mrow" id="MathJax-Span-1924"><span class="mn" id="MathJax-Span-1925" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1935"><span class="mrow" id="MathJax-Span-1936"><span class="mn" id="MathJax-Span-1937" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span><span style="position: absolute; clip: rect(2.979em, 1000.48em, 8.217em, -999.997em); top: -5.89em; left: 6.848em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -6.009em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1901"><span class="mrow" id="MathJax-Span-1902"><span class="mn" id="MathJax-Span-1903" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -4.64em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1914"><span class="mrow" id="MathJax-Span-1915"><span class="mn" id="MathJax-Span-1916" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.211em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1926"><span class="mrow" id="MathJax-Span-1927"><span class="mn" id="MathJax-Span-1928" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1938"><span class="mrow" id="MathJax-Span-1939"><span class="mn" id="MathJax-Span-1940" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.896em;"></span></span></span></span><span class="mo" id="MathJax-Span-1941" style="vertical-align: 2.86em;"><span style="display: inline-block; position: relative; width: 0.301em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.211em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: 0.955em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -2.378em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -1.545em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -0.711em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: 0.122em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span class="mo" id="MathJax-Span-1942" style="font-family: MathJax_Main; padding-left: 0.241em;">∗</span><span class="mrow" id="MathJax-Span-1943" style="padding-left: 0.241em;"><span class="mo" id="MathJax-Span-1944" style="vertical-align: 2.146em;"><span style="display: inline-block; position: relative; width: 0.301em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.211em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -0.414em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -2.318em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -1.366em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mtable" id="MathJax-Span-1945" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 3.515em; height: 0px;"><span style="position: absolute; clip: rect(2.265em, 1000.48em, 6.134em, -999.997em); top: -4.461em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -5.354em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1946"><span class="mrow" id="MathJax-Span-1947"><span class="mn" id="MathJax-Span-1948" style="font-family: MathJax_Main;">2</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.926em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1955"><span class="mrow" id="MathJax-Span-1956"><span class="mn" id="MathJax-Span-1957" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -2.557em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1964"><span class="mrow" id="MathJax-Span-1965"><span class="mn" id="MathJax-Span-1966" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.467em;"></span></span><span style="position: absolute; clip: rect(2.265em, 1000.48em, 6.134em, -999.997em); top: -4.461em; left: 1.491em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -5.354em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1949"><span class="mrow" id="MathJax-Span-1950"><span class="mn" id="MathJax-Span-1951" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.42em, 4.17em, -999.997em); top: -3.926em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1958"><span class="mrow" id="MathJax-Span-1959"><span class="mn" id="MathJax-Span-1960" style="font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -2.557em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1967"><span class="mrow" id="MathJax-Span-1968"><span class="mn" id="MathJax-Span-1969" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.467em;"></span></span><span style="position: absolute; clip: rect(2.265em, 1000.48em, 6.134em, -999.997em); top: -4.461em; left: 2.979em;"><span style="display: inline-block; position: relative; width: 0.479em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -5.354em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1952"><span class="mrow" id="MathJax-Span-1953"><span class="mn" id="MathJax-Span-1954" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -3.926em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1961"><span class="mrow" id="MathJax-Span-1962"><span class="mn" id="MathJax-Span-1963" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -2.557em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1970"><span class="mrow" id="MathJax-Span-1971"><span class="mn" id="MathJax-Span-1972" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.467em;"></span></span></span></span><span class="mo" id="MathJax-Span-1973" style="vertical-align: 2.146em;"><span style="display: inline-block; position: relative; width: 0.301em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.211em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -0.414em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -2.318em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -1.366em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span class="mo" id="MathJax-Span-1974" style="font-family: MathJax_Main; padding-left: 0.241em;">∗</span><span class="mrow" id="MathJax-Span-1975" style="padding-left: 0.241em;"><span class="mo" id="MathJax-Span-1976" style="vertical-align: 2.146em;"><span style="display: inline-block; position: relative; width: 0.301em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.211em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -0.414em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -2.318em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -1.366em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mtable" id="MathJax-Span-1977" style="padding-right: 0.182em; padding-left: 0.182em;"><span style="display: inline-block; position: relative; width: 4.289em; height: 0px;"><span style="position: absolute; clip: rect(2.265em, 1001.19em, 6.134em, -999.997em); top: -4.461em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.253em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.48em, 4.17em, -999.997em); top: -5.354em; left: 50%; margin-left: -0.235em;"><span class="mtd" id="MathJax-Span-1978"><span class="mrow" id="MathJax-Span-1979"><span class="mn" id="MathJax-Span-1980" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; 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mode=display" id="MathJax-Element-97">
\begin{vmatrix}
0 & -1.6 & 0.6 \\
0 & 1.2  & 0.8 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{vmatrix} =
\begin{vmatrix}
0.8 & 0.6 & 0 & 0 \\
-0.6 & 0.8 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{vmatrix} *
\begin{vmatrix}
2 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0 \\
\end{vmatrix} *
\begin{vmatrix}
0 & 0 & 1 \\
-1 & 0 & 0 \\
0 & 1 & 0 \\
\end{vmatrix}</script><ul>
<li>上述分解中会构建出一个矩阵∑，该矩阵只有对角元素，其他元素均为0(近似于0)。另一个惯例就是，∑的对角元素是从大到小排列的。这些对角元素称为奇异值。</li>
<li class=" has-jax">奇异值与特征值(PCA 数据中重要特征)是有关系的。这里的奇异值就是矩阵 <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-98-Frame" tabindex="0" style="position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;&amp;#x2217;&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2007" style="width: 7.324em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.074em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.61em, 1006.07em, 2.801em, -999.997em); top: -2.616em; left: 0em;"><span class="mrow" id="MathJax-Span-2008"><span class="mi" id="MathJax-Span-2009" style="font-family: MathJax_Math-italic;">D</span><span class="mi" id="MathJax-Span-2010" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-2011" style="font-family: MathJax_Math-italic;">t</span><span class="mi" id="MathJax-Span-2012" style="font-family: MathJax_Math-italic;">a</span><span class="mo" id="MathJax-Span-2013" style="font-family: MathJax_Main; padding-left: 0.241em;">∗</span><span class="mi" id="MathJax-Span-2014" style="font-family: MathJax_Math-italic; padding-left: 0.241em;">D</span><span class="mi" id="MathJax-Span-2015" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-2016" style="font-family: MathJax_Math-italic;">t</span><span class="msubsup" id="MathJax-Span-2017"><span style="display: inline-block; position: relative; width: 1.134em; height: 0px;"><span style="position: absolute; clip: rect(3.396em, 1000.54em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-2018" style="font-family: MathJax_Math-italic;">a</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -4.342em; left: 0.539em;"><span class="mi" id="MathJax-Span-2019" style="font-size: 70.7%; font-family: MathJax_Math-italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.622em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left: 0px solid; width: 0px; height: 1.146em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mi>a</mi><mi>t</mi><mi>a</mi><mo>∗</mo><mi>D</mi><mi>a</mi><mi>t</mi><msup><mi>a</mi><mi>T</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-98">Data * Data^T</script> 特征值的平方根。</li>
<li>普遍的事实：在某个奇异值的数目(r 个=&gt;奇异值的平方和累加到总值的90%以上)之后，其他的奇异值都置为0(近似于0)。这意味着数据集中仅有 r 个重要特征，而其余特征则都是噪声或冗余特征。</li>
</ul>
<h2 id="SVD-算法特点"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#SVD-%E7%AE%97%E6%B3%95%E7%89%B9%E7%82%B9" class="headerlink" title="SVD 算法特点"></a>SVD 算法特点</h2><p>优点：简化数据，去除噪声，优化算法的结果<br>缺点：数据的转换可能难以理解<br>使用的数据类型：数值型数据</p>
<hr>
<h1 id="推荐系统"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E6%8E%A8%E8%8D%90%E7%B3%BB%E7%BB%9F" class="headerlink" title="推荐系统"></a>推荐系统</h1><p>推荐系统是利用电子商务网站向客户提供商品信息和建议，帮助用户决定应该购买什么产品，模拟销售人员帮助客户完成购买过程。</p>
<h2 id="推荐系统场景"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E6%8E%A8%E8%8D%90%E7%B3%BB%E7%BB%9F%E5%9C%BA%E6%99%AF" class="headerlink" title="推荐系统场景"></a>推荐系统场景</h2><ol>
<li>Amazon 会根据顾客的购买历史向他们推荐物品</li>
<li>Netflix 会向其用户推荐电影</li>
<li>新闻网站会对用户推荐新闻频道</li>
</ol>
<h2 id="推荐系统要点"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E6%8E%A8%E8%8D%90%E7%B3%BB%E7%BB%9F%E8%A6%81%E7%82%B9" class="headerlink" title="推荐系统要点"></a>推荐系统要点</h2><blockquote>
<p>基于协同过滤(collaborative filtering) 的推荐引擎</p>
</blockquote>
<ul>
<li>利用Python 实现 SVD(Numpy 有一个称为 linalg 的线性代数工具箱)</li>
<li>协同过滤：是通过将用户和其他用户的数据进行对比来实现推荐的。</li>
<li>当知道了两个用户或两个物品之间的相似度，我们就可以利用已有的数据来预测未知用户的喜好。</li>
</ul>
<blockquote>
<p>基于物品的相似度和基于用户的相似度：物品比较少则选择物品相似度，用户比较少则选择用户相似度。【矩阵还是小一点好计算】</p>
</blockquote>
<ul>
<li>基于物品的相似度：计算物品之间的距离。【耗时会随物品数量的增加而增加】</li>
<li>由于物品A和物品C 相似度(相关度)很高，所以给买A的人推荐C。</li>
</ul>
<p>用户/物品|物品A|物品B|物品C</p>
<p><img src="https://i.imgur.com/i4FLivm.png" alt="" data-bd-imgshare-binded="1"></p>
<ul>
<li>基于用户的相似度：计算用户之间的距离。【耗时会随用户数量的增加而增加】</li>
<li>由于用户A和用户C 相似度(相关度)很高，所以A和C是兴趣相投的人，对于C买的物品就会推荐给A。</li>
</ul>
<p><img src="https://i.imgur.com/wJ9VndG.png" alt="" data-bd-imgshare-binded="1"></p>
<h2 id="相似度计算"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E7%9B%B8%E4%BC%BC%E5%BA%A6%E8%AE%A1%E7%AE%97" class="headerlink" title="相似度计算"></a>相似度计算</h2><blockquote>
<p>inA, inB 对应的是 列向量</p>
</blockquote>
<ol>
<li>欧氏距离：指在m维空间中两个点之间的真实距离，或者向量的自然长度（即该点到原点的距离）。二维或三维中的欧氏距离就是两点之间的实际距离。<ul>
<li>相似度= 1/(1+欧式距离)</li>
<li><code>相似度= 1.0/(1.0 + la.norm(inA - inB))</code></li>
<li>物品对越相似，它们的相似度值就越大。</li>
</ul>
</li>
<li>皮尔逊相关系数：度量的是两个向量之间的相似度。<ul>
<li>相似度= 0.5 + 0.5<em>corrcoef() 【皮尔逊相关系数的取值范围从 -1 到 +1，通过函数0.5 + 0.5\</em>corrcoef()这个函数计算，把值归一化到0到1之间】</li>
<li><code>相似度= 0.5 + 0.5 * corrcoef(inA, inB, rowvar = 0)[0][1]</code></li>
<li>相对欧氏距离的优势：它对用户评级的量级并不敏感。</li>
</ul>
</li>
<li>余弦相似度：计算的是两个向量夹角的余弦值。<ul>
<li>余弦值 = (A·B)/(||A||·||B||) 【余弦值的取值范围也在-1到+1之间】</li>
<li>相似度= 0.5 + 0.5*余弦值</li>
<li><code>相似度= 0.5 + 0.5*( float(inA.T*inB) / la.norm(inA)*la.norm(inB))</code></li>
<li>如果夹角为90度，则相似度为0；如果两个向量的方向相同，则相似度为1.0。</li>
</ul>
</li>
</ol>
<blockquote>
<p>代码实现</p>
</blockquote>
<pre>'''基于欧氏距离相似度计算，假定inA和inB 都是列向量
相似度=1/(1+距离),相似度介于0-1之间
norm：范式计算，默认是2范数，即:sqrt(a^2+b^2+...)
'''
def ecludSim(inA, inB):
    return 1.0/(1.0 + la.norm(inA - inB))


'''皮尔逊相关系数
范围[-1, 1]，归一化后[0, 1]即0.5 + 0.5 *
相对于欧式距离，对具体量级（五星三星都一样）不敏感皮尔逊相关系数
'''
def pearsSim(inA, inB):
    # 检查是否存在3个或更多的点不存在，该函数返回1.0，此时两个向量完全相关。
    if len(inA) &lt; 3:
        return 1.0
    return 0.5 + 0.5 * corrcoef(inA, inB, rowvar=0)[0][1]


'''计算余弦相似度
如果夹角为90度相似度为0；两个向量的方向相同，相似度为1.0
余弦取值-1到1之间，归一化到0与1之间即：相似度=0.5 + 0.5*cosθ
余弦相似度cosθ=(A*B/|A|*|B|)
'''
def cosSim(inA, inB):
    num = float(inA.T*inB) # 矩阵相乘
    denom = la.norm(inA)*la.norm(inB) # 默认是2范数
    return 0.5 + 0.5*(num/denom)
</pre>

<h2 id="推荐系统的评价"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E6%8E%A8%E8%8D%90%E7%B3%BB%E7%BB%9F%E7%9A%84%E8%AF%84%E4%BB%B7" class="headerlink" title="推荐系统的评价"></a>推荐系统的评价</h2><ul>
<li>采用交叉测试的方法。【拆分数据为训练集和测试集】</li>
<li>推荐引擎评价的指标： 最小均方根误差(Root mean squared error, RMSE)，也称标准误差(Standard error)，就是计算均方误差的平均值然后取其平方根。<ul>
<li>如果RMSE=1, 表示相差1个星级；如果RMSE=2.5, 表示相差2.5个星级。</li>
</ul>
</li>
</ul>
<h2 id="推荐系统原理"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E6%8E%A8%E8%8D%90%E7%B3%BB%E7%BB%9F%E5%8E%9F%E7%90%86" class="headerlink" title="推荐系统原理"></a>推荐系统原理</h2><ul>
<li>推荐系统的工作过程：给定一个用户，系统会为此用户返回N个最好的推荐菜。</li>
<li>实现流程大致如下：<ol>
<li>寻找用户没有评级的菜肴，即在用户-物品矩阵中的0值。</li>
<li>在用户没有评级的所有物品中，对每个物品预计一个可能的评级分数。这就是说：我们认为用户可能会对物品的打分（这就是相似度计算的初衷）。</li>
<li>对这些物品的评分从高到低进行排序，返回前N个物品。</li>
</ol>
</li>
</ul>
<hr>
<h1 id="项目实战-餐馆菜肴推荐系统"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E9%A1%B9%E7%9B%AE%E5%AE%9E%E6%88%98-%E9%A4%90%E9%A6%86%E8%8F%9C%E8%82%B4%E6%8E%A8%E8%8D%90%E7%B3%BB%E7%BB%9F" class="headerlink" title="项目实战: 餐馆菜肴推荐系统"></a>项目实战: 餐馆菜肴推荐系统</h1><p>假如一个人在家决定外出吃饭，但是他并不知道该到哪儿去吃饭，该点什么菜。推荐系统可以帮他做到这两点。</p>
<h2 id="收集并准备数据"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E6%94%B6%E9%9B%86%E5%B9%B6%E5%87%86%E5%A4%87%E6%95%B0%E6%8D%AE" class="headerlink" title="收集并准备数据"></a>收集并准备数据</h2><p><img src="https://i.imgur.com/2ah3bx3.jpg" alt="" data-bd-imgshare-binded="1"></p>
<blockquote>
<p>数据准备的代码实现</p>
</blockquote>
<pre># 利用SVD提高推荐效果，菜肴矩阵
"""
行：代表人
列：代表菜肴名词
值：代表人对菜肴的评分，0表示未评分
"""
def loadExData3():
    return[[2, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5],
           [0, 0, 0, 0, 0, 0, 0, 1, 0, 4, 0],
           [3, 3, 4, 0, 3, 0, 0, 2, 2, 0, 0],
           [5, 5, 5, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 5, 0, 0, 5, 0],
           [4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 5],
           [0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4],
           [0, 0, 0, 0, 0, 0, 5, 0, 0, 5, 0],
           [0, 0, 0, 3, 0, 0, 0, 0, 4, 5, 0],
           [1, 1, 2, 1, 1, 2, 1, 0, 4, 5, 0]]
```

## 分析数据

这里不做过多的讨论(当然此处可以对比不同距离之间的差别)，通常保留矩阵 80% ～ 90% 的能量，就可以得到重要的特征并去除噪声。

```python
'''分析 Sigma 的长度取值
根据自己的业务情况，就行处理，设置对应的 Singma 次数
通常保留矩阵 80% ～ 90% 的能量，就可以得到重要的特征并取出噪声。
'''
def analyse_data(Sigma, loopNum=20):
    # 总方差的集合（总能量值）
    Sig2 = Sigma**2
    SigmaSum = sum(Sig2)
    for i in range(loopNum):
        SigmaI = sum(Sig2[:i+1])
        print('主成分：%s, 方差占比：%s%%' % (format(i+1, '2.0f'), format(SigmaI/SigmaSum*100, '.2f')))
</pre>

<h2 id="训练算法-通过调用-recommend-函数进行推荐"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E8%AE%AD%E7%BB%83%E7%AE%97%E6%B3%95-%E9%80%9A%E8%BF%87%E8%B0%83%E7%94%A8-recommend-%E5%87%BD%E6%95%B0%E8%BF%9B%E8%A1%8C%E6%8E%A8%E8%8D%90" class="headerlink" title="训练算法: 通过调用 recommend() 函数进行推荐"></a>训练算法: 通过调用 recommend() 函数进行推荐</h2><p>recommend() 会调用 基于物品相似度 或者是 基于SVD，得到推荐的物品评分。</p>
<blockquote>
<p>基于物品相似度</p>
</blockquote>
<p><img src="https://i.imgur.com/LsG6hmy.jpg" alt="" data-bd-imgshare-binded="1"></p>
<blockquote>
<p>基于物品相似度的推荐引擎代码实现</p>
</blockquote>
<pre>'''基于物品相似度的推荐引擎
descripte：计算某用户未评分物品中，以对该物品和其他物品评分的用户的物品相似度，然后进行综合评分
dataMat         训练数据集
user            用户编号
simMeas         相似度计算方法
item            未评分的物品编号
Returns: ratSimTotal/simTotal  评分（0～5之间的值）
'''
def standEst(dataMat, user, simMeas, item):
    # 得到数据集中的物品数目
    n = shape(dataMat)[1]
    # 初始化两个评分值
    simTotal = 0.0 ; ratSimTotal = 0.0
    # 遍历行中的每个物品（对用户评过分的物品遍历，并与其他物品进行比较）
    for j in range(n):
        userRating = dataMat[user, j]
        # 如果某个物品的评分值为0，则跳过这个物品
        if userRating == 0: # 终止循环
            continue
        # 寻找两个都评级的物品,变量overLap 给出两个物品中已被评分的元素索引ID
        # logical_and 计算x1和x2元素的真值。
        # print(dataMat[:, item].T.A, ':',dataMat[:, j].T.A )
        overLap = nonzero(logical_and(dataMat[:, item].A &gt; 0, dataMat[:, j].A &gt; 0))[0]
        # 如果相似度为0，则两着没有任何重合元素，终止本次循环
        if len(overLap) == 0:
            similarity = 0
        # 如果存在重合的物品，则基于这些重合物重新计算相似度。
        else:
            similarity = simMeas(dataMat[overLap, item], dataMat[overLap, j])
        # 相似度会不断累加，每次计算时还考虑相似度和当前用户评分的乘积
        # similarity  用户相似度，   userRating 用户评分
        simTotal += similarity
        ratSimTotal += similarity * userRating
    if simTotal == 0:
        return 0
    # 通过除以所有的评分总和，对上述相似度评分的乘积进行归一化，使得最后评分在0~5之间，这些评分用来对预测值进行排序
    else:
        return ratSimTotal/simTotal
</pre>

<blockquote>
<p>基于SVD</p>
</blockquote>
<p><img src="https://i.imgur.com/sBd8RsA.png" alt="" data-bd-imgshare-binded="1"></p>
<blockquote>
<p>基于SVD的代码实现</p>
</blockquote>
<pre>'''分析 Sigma 的长度取值
根据自己的业务情况，就行处理，设置对应的 Singma 次数
通常保留矩阵 80% ～ 90% 的能量，就可以得到重要的特征并取出噪声。
'''
def analyse_data(Sigma, loopNum=20):
    # 总方差的集合（总能量值）
    Sig2 = Sigma**2
    SigmaSum = sum(Sig2)
    for i in range(loopNum):
        SigmaI = sum(Sig2[:i+1])
        print('主成分：%s, 方差占比：%s%%' % (format(i+1, '2.0f'), format(SigmaI/SigmaSum*100, '.2f')))


'''基于SVD的评分估计
Args:
    dataMat         训练数据集
    user            用户编号
    simMeas         相似度计算方法
    item            未评分的物品编号
Returns:
    ratSimTotal/simTotal     评分（0～5之间的值）
'''
def svdEst(dataMat, user, simMeas, item):
    # 物品数目
    n = shape(dataMat)[1]
    # 对数据集进行SVD分解
    simTotal = 0.0 ;  ratSimTotal = 0.0
    # 奇异值分解,只利用90%能量值的奇异值，奇异值以NumPy数组形式保存
    U, Sigma, VT = la.svd(dataMat)
    # 分析 Sigma 的长度取值
    # analyse_data(Sigma, 20)

    # 如果要进行矩阵运算，就必须要用这些奇异值构建出一个对角矩阵
    Sig4 = mat(eye(4) * Sigma[: 4]) # eye对角矩阵
    # 利用U矩阵将物品转换到低维空间中，构建转换后的物品(物品+4个主要的特征)
    xformedItems = dataMat.T * U[:, :4] * Sig4.I # I 逆矩阵
    # print('dataMat', shape(dataMat))
    # print('U[:, :4]', shape(U[:, :4]))
    # print('Sig4.I', shape(Sig4.I))
    # print('VT[:4, :]', shape(VT[:4, :]))
    # print('xformedItems', shape(xformedItems))

    # 对于给定的用户，for循环在用户对应行的元素上进行遍历
    # 和standEst()函数的for循环一样，这里相似度计算在低维空间下进行的。
    for j in range(n):
        userRating = dataMat[user, j]
        if userRating == 0 or j == item:
            continue
        # 相似度的计算方法也会作为一个参数传递给该函数
        similarity = simMeas(xformedItems[item, :].T, xformedItems[j, :].T)
        # for 循环中加入了一条print语句，以便了解相似度计算的进展情况。如果觉得累赘，可以去掉
        # print('the %d and %d similarity is: %f' % (item, j, similarity))
        # 对相似度不断累加求和
        simTotal += similarity
        # 对相似度及对应评分值的乘积求和
        ratSimTotal += similarity * userRating
    if simTotal == 0:
        return 0
    else:
        # 计算估计评分
        return ratSimTotal/simTotal
</pre>

<h2 id="排序获取最后的推荐结果"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E6%8E%92%E5%BA%8F%E8%8E%B7%E5%8F%96%E6%9C%80%E5%90%8E%E7%9A%84%E6%8E%A8%E8%8D%90%E7%BB%93%E6%9E%9C" class="headerlink" title="排序获取最后的推荐结果"></a>排序获取最后的推荐结果</h2><pre>'''recommend函数推荐引擎，默认调用standEst函数，产生最高的N个推荐结果
Args:
    dataMat         训练数据集
    user            用户编号
    simMeas         相似度计算方法
    estMethod       使用的推荐算法
Returns:  返回最终 N 个推荐结果
'''
def recommend(dataMat, user, N=3, simMeas=cosSim, estMethod=standEst):
    # 寻找未评级的物品,对给定的用户建立一个未评分的物品列表
    unratedItems = nonzero(dataMat[user, :].A == 0)[1] # .A: 矩阵转数组
    # 如果不存在未评分物品，那么就退出函数
    if len(unratedItems) == 0:
        return 'you rated everything'
    # 物品的编号和评分值
    itemScores = []
    # 在未评分物品上进行循环
    for item in unratedItems:
        # 获取 item 该物品的评分
        estimatedScore = estMethod(dataMat, user, simMeas, item)
        itemScores.append((item, estimatedScore))
    # 按照评分得分 进行逆排序，获取前N个未评级物品进行推荐
    return sorted(itemScores, key=lambda jj: jj[1], reverse=True)[: N]
</pre>

<h2 id="测试和项目调用"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E6%B5%8B%E8%AF%95%E5%92%8C%E9%A1%B9%E7%9B%AE%E8%B0%83%E7%94%A8" class="headerlink" title="测试和项目调用"></a>测试和项目调用</h2><blockquote>
<p>测试代码</p>
</blockquote>
<pre># 计算相似度的方法
myMat = mat(loadExData3())
# 计算相似度的第一种方式
# print(recommend(myMat, 1, estMethod=svdEst))
# 计算相似度的第二种方式
# print(recommend(myMat, 1, estMethod=svdEst, simMeas=pearsSim))

# 默认推荐（菜馆菜肴推荐示例）
print(recommend(myMat, 2))
</pre>

<blockquote>
<p>运行结果</p>
</blockquote>
<pre><code>菜馆菜肴推荐结果： [(3, 4.0), (5, 4.0), (6, 4.0)]

***Repl Closed***
</code></pre><p>分析结果，我们不难发现，分别对3烤牛肉，5鲁宾三明治、6印度烤鸡给我4星好评，推荐给我们的用户。</p>
<h2 id="要点补充"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E8%A6%81%E7%82%B9%E8%A1%A5%E5%85%85" class="headerlink" title="要点补充"></a>要点补充</h2><blockquote>
<p>基于内容(content-based)的推荐</p>
</blockquote>
<ol>
<li>通过各种标签来标记菜肴</li>
<li>将这些属性作为相似度计算所需要的数据</li>
<li>这就是：基于内容的推荐。</li>
</ol>
<blockquote>
<p>构建推荐引擎面临的挑战</p>
</blockquote>
<p>问题</p>
<ul>
<li>1）在大规模的数据集上，SVD分解会降低程序的速度</li>
<li>2）存在其他很多规模扩展性的挑战性问题，比如矩阵的表示方法和计算相似度得分消耗资源。</li>
<li>3）如何在缺乏数据时给出好的推荐-称为冷启动【简单说：用户不会喜欢一个无效的物品，而用户不喜欢的物品又无效】</li>
</ul>
<p>建议</p>
<ul>
<li>1）在大型系统中，SVD分解(可以在程序调入时运行一次)每天运行一次或者其频率更低，并且还要离线运行。</li>
<li>2）在实际中，另一个普遍的做法就是离线计算并保存相似度得分。(物品相似度可能被用户重复的调用)</li>
<li>3）冷启动问题，解决方案就是将推荐看成是搜索问题，通过各种标签／属性特征进行<code>基于内容的推荐</code>。</li>
</ul>
<hr>
<h1 id="项目案例-基于SVD的图像压缩"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E9%A1%B9%E7%9B%AE%E6%A1%88%E4%BE%8B-%E5%9F%BA%E4%BA%8ESVD%E7%9A%84%E5%9B%BE%E5%83%8F%E5%8E%8B%E7%BC%A9" class="headerlink" title="项目案例: 基于SVD的图像压缩"></a>项目案例: 基于SVD的图像压缩</h1><h2 id="收集并准备数据-1"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E6%94%B6%E9%9B%86%E5%B9%B6%E5%87%86%E5%A4%87%E6%95%B0%E6%8D%AE-1" class="headerlink" title="收集并准备数据"></a>收集并准备数据</h2><p>将文本数据转化为矩阵</p>
<pre>'''图像压缩函数'''
def imgLoadData(filename):
    myl = []
    for line in open(filename).readlines():
        newRow = []
        for i in range(32):
            newRow.append(int(line[i]))
        myl.append(newRow)
    # 矩阵调入后，就可以在屏幕上输出该矩阵
    myMat = mat(myl)
    return myMat
</pre>

<h2 id="分析数据-分析Sigma的长度个数"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E5%88%86%E6%9E%90%E6%95%B0%E6%8D%AE-%E5%88%86%E6%9E%90Sigma%E7%9A%84%E9%95%BF%E5%BA%A6%E4%B8%AA%E6%95%B0" class="headerlink" title="分析数据: 分析Sigma的长度个数"></a>分析数据: 分析Sigma的长度个数</h2><p>通常保留矩阵 80% ～ 90% 的能量，就可以得到重要的特征并去除噪声。</p>
<pre>'''分析 Sigma 的长度取值
根据自己的业务情况，就行处理，设置对应的 Singma 次数
通常保留矩阵 80% ～ 90% 的能量，就可以得到重要的特征并取出噪声。
'''
def analyse_data(Sigma, loopNum=20):
    # 总方差的集合（总能量值）
    Sig2 = Sigma**2
    SigmaSum = sum(Sig2)
    for i in range(loopNum):
        SigmaI = sum(Sig2[:i+1])
        print('主成分：%s, 方差占比：%s%%' % (format(i+1, '2.0f'), format(SigmaI/SigmaSum*100, '.2f')))
</pre>

<h2 id="使用算法-对比使用-SVD-前后的数据差异对比，对于存储大家可以试着写写"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E4%BD%BF%E7%94%A8%E7%AE%97%E6%B3%95-%E5%AF%B9%E6%AF%94%E4%BD%BF%E7%94%A8-SVD-%E5%89%8D%E5%90%8E%E7%9A%84%E6%95%B0%E6%8D%AE%E5%B7%AE%E5%BC%82%E5%AF%B9%E6%AF%94%EF%BC%8C%E5%AF%B9%E4%BA%8E%E5%AD%98%E5%82%A8%E5%A4%A7%E5%AE%B6%E5%8F%AF%E4%BB%A5%E8%AF%95%E7%9D%80%E5%86%99%E5%86%99" class="headerlink" title="使用算法: 对比使用 SVD 前后的数据差异对比，对于存储大家可以试着写写"></a>使用算法: 对比使用 SVD 前后的数据差异对比，对于存储大家可以试着写写</h2><p>例如：<code>32*32=1024 =&gt; 32*2+2*1+32*2=130</code>(2*1表示去掉了除对角线的0), 几乎获得了10倍的压缩比。</p>
<pre>'''打印矩阵
由于矩阵保护了浮点数，因此定义浅色和深色，遍历所有矩阵元素，当元素大于阀值时打印1，否则打印0
'''
def printMat(inMat, thresh=0.8):
    for i in range(32):
        for k in range(32):
            if float(inMat[i, k]) &gt; thresh:
                print(1)
            else:
                print(0)
        print('')


'''实现图像压缩，允许基于任意给定的奇异值数目来重构图像
Args:
    numSV       Sigma长度
    thresh      判断的阈值
'''
def imgCompress(numSV=3, thresh=0.8):
    # 构建一个列表
    myMat = imgLoadData('./0_5.txt')

    print("****original matrix****")
    # 对原始图像进行SVD分解并重构图像e
    printMat(myMat, thresh)

    # 通过Sigma 重新构成SigRecom来实现
    # Sigma是一个对角矩阵，因此需要建立一个全0矩阵，然后将前面的那些奇异值填充到对角线上。
    U, Sigma, VT = la.svd(myMat)
    # SigRecon = mat(zeros((numSV, numSV)))
    # for k in range(numSV):
    #     SigRecon[k, k] = Sigma[k]

    # 分析插入的 Sigma 长度
    # analyse_data(Sigma, 20)

    SigRecon = mat(eye(numSV) * Sigma[: numSV])
    reconMat = U[:, :numSV] * SigRecon * VT[:numSV, :]
    print("****reconstructed matrix using %d singular values *****" % numSV)
    printMat(reconMat, thresh)
</pre>


<h2 id="参考文献"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E5%8F%82%E8%80%83%E6%96%87%E7%8C%AE" class="headerlink" title="参考文献"></a>参考文献</h2><ol>
<li><a href="https://zh.wikipedia.org/zh-hans/%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3" target="_blank" rel="noopener">奇异值分解</a></li>
<li><a href="https://zh.wikipedia.org/wiki/" target="_blank" rel="noopener">中文维基百科</a></li>
<li><a href="https://github.com/BaiNingchao/MachineLearning-1" target="_blank" rel="noopener">GitHub</a></li>
<li>图书：《机器学习实战》</li>
<li><a href="https://baike.baidu.com/item/%E8%87%AA%E7%84%B6%E8%AF%AD%E8%A8%80%E5%A4%84%E7%90%86%E7%90%86%E8%AE%BA%E4%B8%8E%E5%AE%9E%E6%88%98" target="_blank" rel="noopener">图书：《自然语言处理理论与实战》</a></li>
<li><a href="https://www.cnblogs.com/LeftNotEasy/archive/2011/01/19/svd-and-applications.html" target="_blank" rel="noopener">强大的矩阵奇异值分解(SVD)及其应用</a></li>
<li><a href="https://blog.csdn.net/u010099080/article/details/68060274" target="_blank" rel="noopener">奇异值分解 SVD 的数学解释</a></li>
</ol>
<h2 id="完整代码下载"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E5%AE%8C%E6%95%B4%E4%BB%A3%E7%A0%81%E4%B8%8B%E8%BD%BD" class="headerlink" title="完整代码下载"></a>完整代码下载</h2><blockquote>
<p>源码请进【机器学习和自然语言QQ群：436303759】文件下载：<a target="_blank" href="http://shang.qq.com/wpa/qunwpa?idkey=ef3bbb679b06ac59b136c57ba9e7935ff9d3b10faeabde6e4efcafe523bbbf4d"><img border="0" src="./一步步教你轻松学奇异值分解SVD降维算法 _ 白宁超的官网_files/group.png" alt="自然语言处理和机器学习技术QQ交流" title="自然语言处理和机器学习技术交流" data-bd-imgshare-binded="1"></a></p>
</blockquote>
<p><img src="https://i.imgur.com/kIFJ33g.jpg" alt="" data-bd-imgshare-binded="1"></p>
<h2 id="作者声明"><a href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E4%BD%9C%E8%80%85%E5%A3%B0%E6%98%8E" class="headerlink" title="作者声明"></a>作者声明</h2><blockquote>
<p>本文版权归作者所有，旨在技术交流使用。未经作者同意禁止转载，转载后需在文章页面明显位置给出原文连接，否则相关责任自行承担。</p>
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              <div class="post-toc-content"><ol class="nav"><li class="nav-item nav-level-1"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3%E5%8E%9F%E7%90%86"><span class="nav-number">1.</span> <span class="nav-text">奇异值分解原理</span></a><ol class="nav-child"><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E4%BB%80%E4%B9%88%E6%98%AF%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3-SVD%EF%BC%89"><span class="nav-number">1.1.</span> <span class="nav-text">什么是奇异值分解(SVD）</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E7%9F%A9%E9%98%B5%E7%9F%A5%E8%AF%86"><span class="nav-number">1.2.</span> <span class="nav-text">矩阵知识</span></a></li></ol></li><li class="nav-item nav-level-1"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#SVD-%E7%9A%84%E8%AE%A1%E7%AE%97%E6%96%B9%E6%B3%95"><span class="nav-number">2.</span> <span class="nav-text">SVD 的计算方法</span></a><ol class="nav-child"><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#SVD-%E4%B8%8E%E7%89%B9%E5%BE%81%E5%80%BC"><span class="nav-number">2.1.</span> <span class="nav-text">SVD 与特征值</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E5%A6%82%E4%BD%95%E8%AE%A1%E7%AE%97-SVD"><span class="nav-number">2.2.</span> <span class="nav-text">如何计算 SVD</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E5%AE%9E%E9%99%85%E8%AE%A1%E7%AE%97%E7%9C%8B%E7%9C%8B"><span class="nav-number">2.3.</span> <span class="nav-text">实际计算看看</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E5%87%A0%E4%BD%95%E4%B8%8A%E7%9A%84%E7%9B%B4%E8%A7%82%E8%A7%A3%E9%87%8A"><span class="nav-number">2.4.</span> <span class="nav-text">几何上的直观解释</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#SVD-%E5%9C%BA%E6%99%AF"><span class="nav-number">2.5.</span> <span class="nav-text">SVD 场景</span></a></li></ol></li><li class="nav-item nav-level-1"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#SVD-%E5%B7%A5%E4%BD%9C%E5%8E%9F%E7%90%86"><span class="nav-number">3.</span> <span class="nav-text">SVD 工作原理</span></a><ol class="nav-child"><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E7%9F%A9%E9%98%B5%E5%88%86%E8%A7%A3"><span class="nav-number">3.1.</span> <span class="nav-text">矩阵分解</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#SVD-%E6%98%AF%E7%9F%A9%E9%98%B5%E5%88%86%E8%A7%A3%E7%9A%84%E4%B8%80%E7%A7%8D%E7%B1%BB%E5%9E%8B%EF%BC%8C%E4%B9%9F%E6%98%AF%E7%9F%A9%E9%98%B5%E5%88%86%E8%A7%A3%E6%9C%80%E5%B8%B8%E8%A7%81%E7%9A%84%E6%8A%80%E6%9C%AF"><span class="nav-number">3.2.</span> <span class="nav-text">SVD 是矩阵分解的一种类型，也是矩阵分解最常见的技术</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#SVD-%E7%AE%97%E6%B3%95%E7%89%B9%E7%82%B9"><span class="nav-number">3.3.</span> <span class="nav-text">SVD 算法特点</span></a></li></ol></li><li class="nav-item nav-level-1"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E6%8E%A8%E8%8D%90%E7%B3%BB%E7%BB%9F"><span class="nav-number">4.</span> <span class="nav-text">推荐系统</span></a><ol class="nav-child"><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E6%8E%A8%E8%8D%90%E7%B3%BB%E7%BB%9F%E5%9C%BA%E6%99%AF"><span class="nav-number">4.1.</span> <span class="nav-text">推荐系统场景</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E6%8E%A8%E8%8D%90%E7%B3%BB%E7%BB%9F%E8%A6%81%E7%82%B9"><span class="nav-number">4.2.</span> <span class="nav-text">推荐系统要点</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E7%9B%B8%E4%BC%BC%E5%BA%A6%E8%AE%A1%E7%AE%97"><span class="nav-number">4.3.</span> <span class="nav-text">相似度计算</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E6%8E%A8%E8%8D%90%E7%B3%BB%E7%BB%9F%E7%9A%84%E8%AF%84%E4%BB%B7"><span class="nav-number">4.4.</span> <span class="nav-text">推荐系统的评价</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E6%8E%A8%E8%8D%90%E7%B3%BB%E7%BB%9F%E5%8E%9F%E7%90%86"><span class="nav-number">4.5.</span> <span class="nav-text">推荐系统原理</span></a></li></ol></li><li class="nav-item nav-level-1 active"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E9%A1%B9%E7%9B%AE%E5%AE%9E%E6%88%98-%E9%A4%90%E9%A6%86%E8%8F%9C%E8%82%B4%E6%8E%A8%E8%8D%90%E7%B3%BB%E7%BB%9F"><span class="nav-number">5.</span> <span class="nav-text">项目实战: 餐馆菜肴推荐系统</span></a><ol class="nav-child"><li class="nav-item nav-level-2 active active-current"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E6%94%B6%E9%9B%86%E5%B9%B6%E5%87%86%E5%A4%87%E6%95%B0%E6%8D%AE"><span class="nav-number">5.1.</span> <span class="nav-text">收集并准备数据</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E8%AE%AD%E7%BB%83%E7%AE%97%E6%B3%95-%E9%80%9A%E8%BF%87%E8%B0%83%E7%94%A8-recommend-%E5%87%BD%E6%95%B0%E8%BF%9B%E8%A1%8C%E6%8E%A8%E8%8D%90"><span class="nav-number">5.2.</span> <span class="nav-text">训练算法: 通过调用 recommend() 函数进行推荐</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E6%8E%92%E5%BA%8F%E8%8E%B7%E5%8F%96%E6%9C%80%E5%90%8E%E7%9A%84%E6%8E%A8%E8%8D%90%E7%BB%93%E6%9E%9C"><span class="nav-number">5.3.</span> <span class="nav-text">排序获取最后的推荐结果</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E6%B5%8B%E8%AF%95%E5%92%8C%E9%A1%B9%E7%9B%AE%E8%B0%83%E7%94%A8"><span class="nav-number">5.4.</span> <span class="nav-text">测试和项目调用</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E8%A6%81%E7%82%B9%E8%A1%A5%E5%85%85"><span class="nav-number">5.5.</span> <span class="nav-text">要点补充</span></a></li></ol></li><li class="nav-item nav-level-1"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E9%A1%B9%E7%9B%AE%E6%A1%88%E4%BE%8B-%E5%9F%BA%E4%BA%8ESVD%E7%9A%84%E5%9B%BE%E5%83%8F%E5%8E%8B%E7%BC%A9"><span class="nav-number">6.</span> <span class="nav-text">项目案例: 基于SVD的图像压缩</span></a><ol class="nav-child"><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E6%94%B6%E9%9B%86%E5%B9%B6%E5%87%86%E5%A4%87%E6%95%B0%E6%8D%AE-1"><span class="nav-number">6.1.</span> <span class="nav-text">收集并准备数据</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E5%88%86%E6%9E%90%E6%95%B0%E6%8D%AE-%E5%88%86%E6%9E%90Sigma%E7%9A%84%E9%95%BF%E5%BA%A6%E4%B8%AA%E6%95%B0"><span class="nav-number">6.2.</span> <span class="nav-text">分析数据: 分析Sigma的长度个数</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E4%BD%BF%E7%94%A8%E7%AE%97%E6%B3%95-%E5%AF%B9%E6%AF%94%E4%BD%BF%E7%94%A8-SVD-%E5%89%8D%E5%90%8E%E7%9A%84%E6%95%B0%E6%8D%AE%E5%B7%AE%E5%BC%82%E5%AF%B9%E6%AF%94%EF%BC%8C%E5%AF%B9%E4%BA%8E%E5%AD%98%E5%82%A8%E5%A4%A7%E5%AE%B6%E5%8F%AF%E4%BB%A5%E8%AF%95%E7%9D%80%E5%86%99%E5%86%99"><span class="nav-number">6.3.</span> <span class="nav-text">使用算法: 对比使用 SVD 前后的数据差异对比，对于存储大家可以试着写写</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E5%8F%82%E8%80%83%E6%96%87%E7%8C%AE"><span class="nav-number">6.4.</span> <span class="nav-text">参考文献</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E5%AE%8C%E6%95%B4%E4%BB%A3%E7%A0%81%E4%B8%8B%E8%BD%BD"><span class="nav-number">6.5.</span> <span class="nav-text">完整代码下载</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="https://bainingchao.github.io/2018/10/11/%E4%B8%80%E6%AD%A5%E6%AD%A5%E6%95%99%E4%BD%A0%E8%BD%BB%E6%9D%BE%E5%AD%A6%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3SVD%E9%99%8D%E7%BB%B4%E7%AE%97%E6%B3%95/#%E4%BD%9C%E8%80%85%E5%A3%B0%E6%98%8E"><span class="nav-number">6.6.</span> <span class="nav-text">作者声明</span></a></li></ol></li></ol></div>
            

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